diff --git a/cranelift/codegen/src/egraph.rs b/cranelift/codegen/src/egraph.rs
index e12b6fed39..f56a562e6a 100644
--- a/cranelift/codegen/src/egraph.rs
+++ b/cranelift/codegen/src/egraph.rs
@@ -134,6 +134,11 @@ where
     /// - Update the value-to-opt-value map, and update the eclass
     ///   union-find, if we rewrote the value to different form(s).
     pub(crate) fn insert_pure_enode(&mut self, inst: NewOrExistingInst) -> Value {
+        match inst {
+            NewOrExistingInst::New(mut data, _) => data.normalize_in_place(),
+            NewOrExistingInst::Existing(inst) => self.func.dfg.insts[inst].normalize_in_place(),
+        }
+
         // Create the external context for looking up and updating the
         // GVN map. This is necessary so that instructions themselves
         // do not have to carry all the references or data for a full
diff --git a/cranelift/codegen/src/ir/condcodes.rs b/cranelift/codegen/src/ir/condcodes.rs
index 7059ce6c92..bdc17c30f4 100644
--- a/cranelift/codegen/src/ir/condcodes.rs
+++ b/cranelift/codegen/src/ir/condcodes.rs
@@ -32,7 +32,7 @@ pub trait CondCode: Copy {
 /// This condition code is used by the `icmp` instruction to compare integer values. There are
 /// separate codes for comparing the integers as signed or unsigned numbers where it makes a
 /// difference.
-#[derive(Clone, Copy, PartialEq, Eq, Debug, Hash)]
+#[derive(Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Debug, Hash)]
 #[cfg_attr(feature = "enable-serde", derive(Serialize, Deserialize))]
 pub enum IntCC {
     /// `==`.
@@ -194,7 +194,7 @@ impl FromStr for IntCC {
 /// The condition codes described here are used to produce a single boolean value from the
 /// comparison. The 14 condition codes here cover every possible combination of the relation above
 /// except the impossible `!UN & !EQ & !LT & !GT` and the always true `UN | EQ | LT | GT`.
-#[derive(Clone, Copy, PartialEq, Eq, Debug, Hash)]
+#[derive(Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Debug, Hash)]
 #[cfg_attr(feature = "enable-serde", derive(Serialize, Deserialize))]
 pub enum FloatCC {
     /// EQ | LT | GT
diff --git a/cranelift/codegen/src/ir/instructions.rs b/cranelift/codegen/src/ir/instructions.rs
index 085f203639..dfecde2bf3 100644
--- a/cranelift/codegen/src/ir/instructions.rs
+++ b/cranelift/codegen/src/ir/instructions.rs
@@ -6,6 +6,7 @@
 //! A large part of this module is auto-generated from the instruction descriptions in the meta
 //! directory.
 
+use crate::ir::condcodes::CondCode;
 use alloc::vec::Vec;
 use core::fmt::{self, Display, Formatter};
 use core::ops::{Deref, DerefMut};
@@ -200,6 +201,32 @@ impl Opcode {
             _ => false,
         }
     }
+
+    /// Returns true if the instruction is commutative: `op(x, y) == op(y, x)` for all x, y
+    pub fn is_commutative(&self) -> bool {
+        match self {
+            Opcode::Band | Opcode::Bor | Opcode::Bxor | Opcode::BxorNot => true,
+
+            Opcode::Iadd
+            | Opcode::Imul
+            | Opcode::IaddPairwise
+            | Opcode::Umulhi
+            | Opcode::Smulhi
+            | Opcode::UaddOverflowTrap
+            | Opcode::Umin
+            | Opcode::Umax
+            | Opcode::Smin
+            | Opcode::Smax
+            | Opcode::AvgRound
+            | Opcode::UaddSat
+            | Opcode::SaddSat => true,
+
+            Opcode::Fadd | Opcode::Fmul => true,
+            Opcode::Fmin | Opcode::Fmax => true,
+
+            _ => false,
+        }
+    }
 }
 
 // This trait really belongs in cranelift-reader where it is used by the `.clif` file parser, but since
@@ -469,6 +496,37 @@ impl InstructionData {
             _ => {}
         }
     }
+
+    /// Normalize commutative instructions by sorting arguments
+    #[inline]
+    pub(crate) fn normalize_in_place(&mut self) {
+        match self {
+            InstructionData::Binary { opcode, args }
+                if args[0] > args[1] && opcode.is_commutative() =>
+            {
+                args.swap(0, 1)
+            }
+            InstructionData::Ternary {
+                opcode: Opcode::Fma,
+                args,
+            } if args[0] > args[1] => args.swap(0, 1),
+            InstructionData::IntCompare { args, cond, .. } if args[0] > args[1] => {
+                args.swap(0, 1);
+                *cond = cond.reverse();
+            }
+            InstructionData::IntCompare { args, cond, .. } if args[0] == args[1] => {
+                *cond = std::cmp::min(*cond, cond.reverse());
+            }
+            InstructionData::FloatCompare { args, cond, .. } if args[0] > args[1] => {
+                args.swap(0, 1);
+                *cond = cond.reverse();
+            }
+            InstructionData::FloatCompare { args, cond, .. } if args[0] == args[1] => {
+                *cond = std::cmp::min(*cond, cond.reverse());
+            }
+            _ => {}
+        }
+    }
 }
 
 /// Information about call instructions.
diff --git a/cranelift/codegen/src/simple_gvn.rs b/cranelift/codegen/src/simple_gvn.rs
index 6b09ae96b2..033f66539d 100644
--- a/cranelift/codegen/src/simple_gvn.rs
+++ b/cranelift/codegen/src/simple_gvn.rs
@@ -35,6 +35,14 @@ struct HashKey<'a, 'f: 'a> {
     ty: Type,
     pos: &'a RefCell<FuncCursor<'f>>,
 }
+
+impl<'a, 'f: 'a> HashKey<'a, 'f> {
+    fn new(mut inst: InstructionData, ty: Type, pos: &'a RefCell<FuncCursor<'f>>) -> Self {
+        inst.normalize_in_place();
+        Self { inst, ty, pos }
+    }
+}
+
 impl<'a, 'f: 'a> Hash for HashKey<'a, 'f> {
     fn hash<H: Hasher>(&self, state: &mut H) {
         let pool = &self.pos.borrow().func.dfg.value_lists;
@@ -113,11 +121,7 @@ pub fn do_simple_gvn(func: &mut Function, domtree: &mut DominatorTree) {
             }
 
             let ctrl_typevar = func.dfg.ctrl_typevar(inst);
-            let key = HashKey {
-                inst: func.dfg.insts[inst],
-                ty: ctrl_typevar,
-                pos: &pos,
-            };
+            let key = HashKey::new(func.dfg.insts[inst], ctrl_typevar, &pos);
             use crate::scoped_hash_map::Entry::*;
             match visible_values.entry(key) {
                 Occupied(entry) => {
diff --git a/cranelift/filetests/filetests/egraph/algebraic.clif b/cranelift/filetests/filetests/egraph/algebraic.clif
index ce91d0dff6..deeb3d1bf7 100644
--- a/cranelift/filetests/filetests/egraph/algebraic.clif
+++ b/cranelift/filetests/filetests/egraph/algebraic.clif
@@ -283,8 +283,6 @@ block0(v0: i32):
     v2 = imul v1, v0
     return v2
     ; check: v3 = ineg v0
-    ; check: v5 -> v3
-    ; check: v6 -> v3
     ; check: return v3
 }
 
@@ -326,8 +324,8 @@ block0(v0: i8):
     v3 = iconst.i8 3
     v4 = bor v3, v2
     return v4
-    ; check: v6 = bor v0, v3
-    ; check: return v6
+    ; check: v5 = bor v0, v3  ; v3 = 3
+    ; check: return v5
 }
 
 function %or_and_constant_with_any_constant_should_not_apply_rule_i8(i8) -> i8 {
@@ -379,8 +377,8 @@ block0(v0: i64):
     v3 = iconst.i64 3
     v4 = bor v3, v2
     return v4
-    ; check: v6 = bor v0, v3
-    ; check: return v6
+    ; check: v5 = bor v0, v3  ; v3 = 3
+    ; check: return v5
 }
 
 function %or_and_constant_with_any_constant_should_not_apply_rule_i64(i64) -> i64 {
@@ -421,8 +419,8 @@ block0(v1: i64):
     return v3
 }
 
-; check: v5 = bnot v1
-; check: return v5
+; check: v4 = bnot v1
+; check: return v4
 
 function %extend_always_above_zero(i32) -> i8 {
 block0(v1: i32):
diff --git a/cranelift/filetests/filetests/egraph/basic-gvn.clif b/cranelift/filetests/filetests/egraph/basic-gvn.clif
index 3d74a31b1e..a4a6912e47 100644
--- a/cranelift/filetests/filetests/egraph/basic-gvn.clif
+++ b/cranelift/filetests/filetests/egraph/basic-gvn.clif
@@ -22,7 +22,7 @@ block2(v6: i32):
 ; check: block0(v0: i32, v1: i32):
 ; check:      v2 = iadd v0, v1
 ; check:  block1:
-; check:      v5 = iadd.i32 v2, v0
+; check:      v5 = iadd.i32 v0, v2
 ; nextln:     return v5
 ; check: block2:
 ; nextln:    return v1
diff --git a/cranelift/filetests/filetests/egraph/load-hoist.clif b/cranelift/filetests/filetests/egraph/load-hoist.clif
index f52a1995c4..a802a6ff96 100644
--- a/cranelift/filetests/filetests/egraph/load-hoist.clif
+++ b/cranelift/filetests/filetests/egraph/load-hoist.clif
@@ -39,6 +39,6 @@ function %foo(i64 vmctx, i64, i32, i32) -> i32 fast {
 
 ; check:    v9 = load.i64 notrap aligned readonly v0+80
 ; check: block2(v6: i32, v7: i32, v15: i32):
-; check:    v10 = iadd.i64 v9, v8
+; check:    v10 = iadd v8, v9
 ; check:    v11 = load.i32 little heap v10
 ; check:    brif v17, block2(v12, v14, v17), block4
diff --git a/cranelift/filetests/filetests/simple_gvn/commutative.clif b/cranelift/filetests/filetests/simple_gvn/commutative.clif
new file mode 100644
index 0000000000..760d395e92
--- /dev/null
+++ b/cranelift/filetests/filetests/simple_gvn/commutative.clif
@@ -0,0 +1,55 @@
+test simple-gvn
+
+function %commutative_binary(i32, i32) -> i32, i32 {
+block0(v0: i32, v1: i32):
+    v2 = iadd v0, v1
+    v3 = iadd v1, v0
+    return v2, v3
+    ; check: v2 = iadd v0, v1
+    ; check: return v2, v2
+}
+
+function %commutative_ternary(f32, f32, f32) -> f32, f32 {
+block0(v0: f32, v1: f32, v2: f32):
+    v3 = fma v0, v1, v2
+    v4 = fma v1, v0, v2
+    return v3, v4
+    ; check: v3 = fma v0, v1, v2
+    ; check: return v3, v3
+}
+
+function %commutative_icmp(i32, i32) -> i8, i8 {
+block0(v0: i32, v1: i32):
+    v2 = icmp ult v0, v1
+    v3 = icmp ugt v1, v0
+    return v2, v3
+    ; check: v2 = icmp ult v0, v1
+    ; check: return v2, v2
+}
+
+function %commutative_icmp_reflexive(i32, i32) -> i8, i8 {
+block0(v0: i32, v1: i32):
+    v2 = icmp ule v0, v0
+    v3 = icmp uge v0, v0
+    return v2, v3
+    ; check: v2 = icmp ule v0, v0
+    ; check: return v2, v2
+}
+
+function %commutative_fcmp(f32, f32) -> i8, i8 {
+block0(v0: f32, v1: f32):
+    v2 = fcmp lt v0, v1
+    v3 = fcmp gt v1, v0
+    return v2, v3
+    ; check: v2 = fcmp lt v0, v1
+    ; check: return v2, v2
+}
+
+function %commutative_fcmp_reflexive(f32, f32) -> i8, i8 {
+block0(v0: f32, v1: f32):
+    v2 = fcmp ule v0, v0
+    v3 = fcmp uge v0, v0
+    return v2, v3
+    ; check: v2 = fcmp ule v0, v0
+    ; check: return v2, v2
+}
diff --git a/cranelift/filetests/filetests/wasm/duplicate-loads-dynamic-memory-egraph.wat b/cranelift/filetests/filetests/wasm/duplicate-loads-dynamic-memory-egraph.wat
index 73aaccd829..685fd3cd51 100644
--- a/cranelift/filetests/filetests/wasm/duplicate-loads-dynamic-memory-egraph.wat
+++ b/cranelift/filetests/filetests/wasm/duplicate-loads-dynamic-memory-egraph.wat
@@ -54,7 +54,7 @@
 ;; @0057                               v4 = uextend.i64 v0
 ;; @0057                               v6 = icmp ugt v4, v5
 ;; @0057                               v9 = iconst.i64 0
-;; @0057                               v8 = iadd v7, v4
+;; @0057                               v8 = iadd v4, v7
 ;; @0057                               v10 = select_spectre_guard v6, v9, v8  ; v9 = 0
 ;; @0057                               v11 = load.i32 little heap v10
 ;;                                     v2 -> v11
@@ -75,7 +75,7 @@
 ;; @0064                               v4 = uextend.i64 v0
 ;; @0064                               v6 = icmp ugt v4, v5
 ;; @0064                               v10 = iconst.i64 0
-;; @0064                               v8 = iadd v7, v4
+;; @0064                               v8 = iadd v4, v7
 ;;                                     v22 = iconst.i64 1234
 ;; @0064                               v9 = iadd v8, v22  ; v22 = 1234
 ;; @0064                               v11 = select_spectre_guard v6, v10, v9  ; v10 = 0
diff --git a/cranelift/filetests/filetests/wasm/duplicate-loads-static-memory-egraph.wat b/cranelift/filetests/filetests/wasm/duplicate-loads-static-memory-egraph.wat
index d434d5a33a..1a8571eff6 100644
--- a/cranelift/filetests/filetests/wasm/duplicate-loads-static-memory-egraph.wat
+++ b/cranelift/filetests/filetests/wasm/duplicate-loads-static-memory-egraph.wat
@@ -44,9 +44,9 @@
 ;;     gv1 = load.i64 notrap aligned readonly gv0
 ;;
 ;;                                 block0(v0: i32, v1: i64):
-;; @0057                               v5 = load.i64 notrap aligned readonly v1
 ;; @0057                               v4 = uextend.i64 v0
-;; @0057                               v6 = iadd v5, v4
+;; @0057                               v5 = load.i64 notrap aligned readonly v1
+;; @0057                               v6 = iadd v4, v5
 ;; @0057                               v7 = load.i32 little heap v6
 ;;                                     v2 -> v7
 ;; @005f                               jump block1
@@ -60,9 +60,9 @@
 ;;     gv1 = load.i64 notrap aligned readonly gv0
 ;;
 ;;                                 block0(v0: i32, v1: i64):
-;; @0064                               v5 = load.i64 notrap aligned readonly v1
 ;; @0064                               v4 = uextend.i64 v0
-;; @0064                               v6 = iadd v5, v4
+;; @0064                               v5 = load.i64 notrap aligned readonly v1
+;; @0064                               v6 = iadd v4, v5
 ;;                                     v14 = iconst.i64 1234
 ;; @0064                               v7 = iadd v6, v14  ; v14 = 1234
 ;; @0064                               v8 = load.i32 little heap v7
diff --git a/cranelift/filetests/filetests/wasm/dynamic-memory-no-spectre-access-same-index-different-offsets.wat b/cranelift/filetests/filetests/wasm/dynamic-memory-no-spectre-access-same-index-different-offsets.wat
index 14da1eb483..50894bda71 100644
--- a/cranelift/filetests/filetests/wasm/dynamic-memory-no-spectre-access-same-index-different-offsets.wat
+++ b/cranelift/filetests/filetests/wasm/dynamic-memory-no-spectre-access-same-index-different-offsets.wat
@@ -72,7 +72,7 @@
 ;;
 ;;                                 block3:
 ;; @0047                               v8 = load.i64 notrap aligned v1
-;; @0047                               v9 = iadd v8, v5
+;; @0047                               v9 = iadd.i64 v5, v8
 ;; @0047                               v10 = load.i32 little heap v9
 ;;                                     v2 -> v10
 ;; @004c                               brif.i8 v7, block4, block5
@@ -87,7 +87,7 @@
 ;;                                     v3 -> v17
 ;; @0051                               v19 = iconst.i64 0x0010_0003
 ;; @0051                               v20 = uadd_overflow_trap.i64 v5, v19, heap_oob  ; v19 = 0x0010_0003
-;; @0051                               v22 = icmp ugt v20, v6
+;; @0051                               v22 = icmp.i64 ult v6, v20
 ;; @0051                               brif v22, block6, block7
 ;;
 ;;                                 block6 cold:
@@ -95,7 +95,7 @@
 ;;
 ;;                                 block7:
 ;; @0051                               v23 = load.i64 notrap aligned v1
-;; @0051                               v24 = iadd v23, v5
+;; @0051                               v24 = iadd.i64 v5, v23
 ;;                                     v28 = iconst.i64 0x000f_ffff
 ;; @0051                               v25 = iadd v24, v28  ; v28 = 0x000f_ffff
 ;; @0051                               v26 = load.i32 little heap v25
@@ -122,7 +122,7 @@
 ;;
 ;;                                 block3:
 ;; @005d                               v8 = load.i64 notrap aligned v4
-;; @005d                               v9 = iadd v8, v5
+;; @005d                               v9 = iadd.i64 v5, v8
 ;; @005d                               store.i32 little heap v1, v9
 ;; @0064                               brif.i8 v7, block4, block5
 ;;
@@ -135,7 +135,7 @@
 ;; @0064                               store.i32 little heap v2, v15
 ;; @006b                               v17 = iconst.i64 0x0010_0003
 ;; @006b                               v18 = uadd_overflow_trap.i64 v5, v17, heap_oob  ; v17 = 0x0010_0003
-;; @006b                               v20 = icmp ugt v18, v6
+;; @006b                               v20 = icmp.i64 ult v6, v18
 ;; @006b                               brif v20, block6, block7
 ;;
 ;;                                 block6 cold:
@@ -143,7 +143,7 @@
 ;;
 ;;                                 block7:
 ;; @006b                               v21 = load.i64 notrap aligned v4
-;; @006b                               v22 = iadd v21, v5
+;; @006b                               v22 = iadd.i64 v5, v21
 ;;                                     v25 = iconst.i64 0x000f_ffff
 ;; @006b                               v23 = iadd v22, v25  ; v25 = 0x000f_ffff
 ;; @006b                               store.i32 little heap v3, v23
diff --git a/cranelift/filetests/filetests/wasm/dynamic-memory-yes-spectre-access-same-index-different-offsets.wat b/cranelift/filetests/filetests/wasm/dynamic-memory-yes-spectre-access-same-index-different-offsets.wat
index 4dbe53e807..66599d66ea 100644
--- a/cranelift/filetests/filetests/wasm/dynamic-memory-yes-spectre-access-same-index-different-offsets.wat
+++ b/cranelift/filetests/filetests/wasm/dynamic-memory-yes-spectre-access-same-index-different-offsets.wat
@@ -67,7 +67,7 @@
 ;; @0047                               v5 = uextend.i64 v0
 ;; @0047                               v7 = icmp ugt v5, v6
 ;; @0047                               v10 = iconst.i64 0
-;; @0047                               v9 = iadd v8, v5
+;; @0047                               v9 = iadd v5, v8
 ;; @0047                               v11 = select_spectre_guard v7, v10, v9  ; v10 = 0
 ;; @0047                               v12 = load.i32 little heap v11
 ;;                                     v2 -> v12
@@ -78,7 +78,7 @@
 ;;                                     v3 -> v21
 ;; @0051                               v23 = iconst.i64 0x0010_0003
 ;; @0051                               v24 = uadd_overflow_trap v5, v23, heap_oob  ; v23 = 0x0010_0003
-;; @0051                               v26 = icmp ugt v24, v6
+;; @0051                               v26 = icmp ult v6, v24
 ;;                                     v34 = iconst.i64 0x000f_ffff
 ;; @0051                               v29 = iadd v9, v34  ; v34 = 0x000f_ffff
 ;; @0051                               v31 = select_spectre_guard v26, v10, v29  ; v10 = 0
@@ -101,7 +101,7 @@
 ;; @005d                               v5 = uextend.i64 v0
 ;; @005d                               v7 = icmp ugt v5, v6
 ;; @005d                               v10 = iconst.i64 0
-;; @005d                               v9 = iadd v8, v5
+;; @005d                               v9 = iadd v5, v8
 ;; @005d                               v11 = select_spectre_guard v7, v10, v9  ; v10 = 0
 ;; @005d                               store little heap v1, v11
 ;;                                     v30 = iconst.i64 4
@@ -110,7 +110,7 @@
 ;; @0064                               store little heap v2, v19
 ;; @006b                               v21 = iconst.i64 0x0010_0003
 ;; @006b                               v22 = uadd_overflow_trap v5, v21, heap_oob  ; v21 = 0x0010_0003
-;; @006b                               v24 = icmp ugt v22, v6
+;; @006b                               v24 = icmp ult v6, v22
 ;;                                     v31 = iconst.i64 0x000f_ffff
 ;; @006b                               v27 = iadd v9, v31  ; v31 = 0x000f_ffff
 ;; @006b                               v29 = select_spectre_guard v24, v10, v27  ; v10 = 0