diff --git a/doc/missed_optimisations.md b/doc/missed_optimisations.md
new file mode 100644
index 000000000000..3e8df3950873
--- /dev/null
+++ b/doc/missed_optimisations.md
@@ -0,0 +1,8 @@
+# List of missed optimisations to look at later
+
+This should be compiled to the identity. But the CSE environment seems to lose
+the relationship in the join after the first switch.
+
+   let f x =
+     let not_x = if x then false else true in
+     if not_x then false else true
diff --git a/flambdatest/mlexamples/involutions.ml b/flambdatest/mlexamples/involutions.ml
new file mode 100644
index 000000000000..3cac5398fc69
--- /dev/null
+++ b/flambdatest/mlexamples/involutions.ml
@@ -0,0 +1,18 @@
+
+let f x =
+  not (not (not x))
+
+let g x =
+  ~- (~- (~- x))
+
+let h x =
+  Int64.(neg (neg (neg x)))
+
+let i x =
+  Int32.(neg (neg (neg x)))
+
+let j x =
+  Nativeint.(neg (neg (neg x)))
+
+let k x =
+  ~-. (~-. (~-. x))
diff --git a/middle_end/flambda/simplify/simplify_primitive.ml b/middle_end/flambda/simplify/simplify_primitive.ml
index 5ecb6ef767a3..d3d898289427 100644
--- a/middle_end/flambda/simplify/simplify_primitive.ml
+++ b/middle_end/flambda/simplify/simplify_primitive.ml
@@ -29,13 +29,16 @@ let apply_cse dacc ~original_prim =
     match DE.find_cse (DA.denv dacc) with_fixed_value with
     | None -> None
     | Some simple ->
-      let canonical =
+      match
         TE.get_canonical_simple_exn (DA.typing_env dacc) simple
           ~min_name_mode:NM.normal
           ~name_mode_of_existing_simple:NM.normal
-      in
-      match canonical with
-      | exception Not_found -> None
+      with
+      | exception Not_found ->
+        (* CR pchambart: this exception was not caught for some time, it is
+           expected never to happen, hence the fatal_error *)
+        Misc.fatal_errorf "No canonical simple for the CSE candidate: %a"
+          Simple.print simple
       | simple -> Some simple
 
 let try_cse dacc ~original_prim ~simplified_args_with_tys ~min_name_mode
@@ -47,12 +50,13 @@ let try_cse dacc ~original_prim ~simplified_args_with_tys ~min_name_mode
   if not (Name_mode.equal min_name_mode Name_mode.normal) then Not_applied dacc
   else
     let result_var = VB.var result_var in
+    let args = List.map fst simplified_args_with_tys in
+    let original_prim = P.update_args original_prim args in
     match apply_cse dacc ~original_prim with
     | Some replace_with ->
       let named = Named.create_simple replace_with in
       let ty = T.alias_type_of (P.result_kind' original_prim) replace_with in
       let env_extension = TEE.one_equation (Name.var result_var) ty in
-      let args = List.map fst simplified_args_with_tys in
       let simplified_named =
         let cost_metrics =
           Cost_metrics.notify_removed
diff --git a/middle_end/flambda/simplify/simplify_unary_primitive.ml b/middle_end/flambda/simplify/simplify_unary_primitive.ml
index c730f6292709..6069c3620e9f 100644
--- a/middle_end/flambda/simplify/simplify_unary_primitive.ml
+++ b/middle_end/flambda/simplify/simplify_unary_primitive.ml
@@ -507,4 +507,14 @@ let simplify_unary_primitive dacc (prim : P.unary_primitive)
   let reachable, env_extension, dacc =
     simplifier dacc ~original_term ~arg ~arg_ty ~result_var
   in
+  let dacc =
+    if P.is_an_involution prim then
+      DA.map_denv dacc ~f:(fun denv ->
+        let prim : P.t =
+          Unary (prim, Simple.var (Var_in_binding_pos.var result_var))
+        in
+        DE.add_cse denv (P.Eligible_for_cse.create_exn prim) ~bound_to:arg)
+    else
+      dacc
+  in
   reachable, env_extension, [arg], dacc
diff --git a/middle_end/flambda/terms/flambda_primitive.ml b/middle_end/flambda/terms/flambda_primitive.ml
index a133f2557234..c95f76f51119 100644
--- a/middle_end/flambda/terms/flambda_primitive.ml
+++ b/middle_end/flambda/terms/flambda_primitive.ml
@@ -944,6 +944,18 @@ let unary_classify_for_printing p =
   | Select_closure _
   | Project_var _ -> Destructive
 
+let is_an_involution (p : unary_primitive) =
+  match p with
+  | Boolean_not -> true
+  | Int_arith (_, Neg) -> true
+  | Float_arith Neg -> true
+    (* Checked through using the following SMT formula with Z3
+       {[ (set-logic QF_FP)
+          (declare-fun x () (_ FloatingPoint 11 53))
+          (assert (not (= x (fp.neg (fp.neg x)))))
+          (check-sat) ]} *)
+  | _ -> false
+
 type binary_int_arith_op =
   | Add | Sub | Mul | Div | Mod | And | Or | Xor
 
@@ -1626,6 +1638,19 @@ let args t =
   | Ternary (_, x0, x1, x2) -> [x0; x1; x2]
   | Variadic (_, xs) -> xs
 
+let update_args t args =
+  match t, args with
+  | Nullary _, [] -> t
+  | Unary (p, _), [x0] -> Unary (p, x0)
+  | Binary (p, _, _), [x0; x1] -> Binary (p, x0, x1)
+  | Ternary (p, _, _, _), [x0; x1; x2] -> Ternary (p, x0, x1, x2)
+  | Variadic (p, l), xs ->
+    assert(List.length l = List.length xs);
+    Variadic (p, xs)
+  | _, _ ->
+    Misc.fatal_errorf "Wrong arity for updating primitive %a with arguments %a"
+      print t Simple.List.print args
+
 let result_kind (t : t) =
   match t with
   | Nullary prim -> result_kind_of_nullary_primitive prim
diff --git a/middle_end/flambda/terms/flambda_primitive.mli b/middle_end/flambda/terms/flambda_primitive.mli
index 240065f22efc..d84468709851 100644
--- a/middle_end/flambda/terms/flambda_primitive.mli
+++ b/middle_end/flambda/terms/flambda_primitive.mli
@@ -334,6 +334,8 @@ include Contains_ids.S with type t := t
 
 val args : t -> Simple.t list
 
+val update_args : t -> Simple.t list -> t
+
 (** Simpler version (e.g. for [Inlining_cost]), where only the actual
     primitive matters, not the arguments. *)
 module Without_args : sig
@@ -404,6 +406,10 @@ val at_most_generative_effects : t -> bool
     and no other effects. *)
 val only_generative_effects : t -> bool
 
+(** Returns [true] iff the primitive is an involution. i.e.
+    x = Prim(y) iff y = Prim(x) *)
+val is_an_involution : unary_primitive -> bool
+
 module Eligible_for_cse : sig
   (** Primitive applications that may be replaced by a variable which is let
       bound to a single instance of such application.  Primitives that are
diff --git a/testsuite/tests/flambda2/involution.ml b/testsuite/tests/flambda2/involution.ml
new file mode 100644
index 000000000000..6138f216bfa8
--- /dev/null
+++ b/testsuite/tests/flambda2/involution.ml
@@ -0,0 +1,45 @@
+(* TEST
+* flambda
+** native
+*)
+
+(* This tests that involutions are effectively detected.
+   It checks that for an involution f, f (f x) == x
+   This is verified by testing that the result of the physical equality
+   is a known constant, through an allocation test. *)
+
+external minor_words : unit -> (float [@unboxed])
+  = "caml_gc_minor_words" "caml_gc_minor_words_unboxed"
+
+let[@inline never] check_no_alloc test_name f x =
+  let before = minor_words () in
+  let _ = Sys.opaque_identity ((f[@inlined never]) x) in
+  let after = minor_words () in
+  let diff = after -. before in
+  if diff = 0. then
+    Format.printf "No allocs for test '%s'@." test_name
+  else
+    Format.printf "Some allocs for test '%s'@." test_name
+
+let () =
+  let[@inline] test_known f =
+    let tester x =
+      let x = Sys.opaque_identity x in
+      let n = if (f[@inlined hint]) ((f[@inlined hint]) x) == x then 1 else 0 in
+      Int64.of_int n
+    in
+    let () = Sys.opaque_identity () in
+    tester
+  in
+
+  (* Test the test: this should allocate *)
+  check_no_alloc "not an involution" (test_known (succ)) 1;
+  (* Actual tests *)
+  check_no_alloc "not" (test_known (not)) true;
+  check_no_alloc "~-." (test_known (~-.)) 42.;
+  check_no_alloc "~-" (test_known (~-)) 42;
+  check_no_alloc "Int32.neg" (test_known Int32.neg) 42l;
+  check_no_alloc "Int64.neg" (test_known Int64.neg) 42L;
+  check_no_alloc "Nativeint.neg" (test_known Nativeint.neg) 42n;
+  ()
+