diff --git a/Project.toml b/Project.toml
index 403da1e65..a56827cad 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRulesCore"
 uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
-version = "1.26.5"
+version = "1.26.6"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
diff --git a/src/projection.jl b/src/projection.jl
index ca577aef2..7ddadf11a 100644
--- a/src/projection.jl
+++ b/src/projection.jl
@@ -452,6 +452,10 @@ end
 (project::ProjectTo{Diagonal})(dx::Diagonal) = Diagonal(project.diag(dx.diag))
 
 # Symmetric
+function _same_uplo(dx::LinearAlgebra.HermOrSym, uplo::Symbol)
+    return LinearAlgebra.sym_uplo(dx.uplo) === uplo
+end
+_same_uplo(dx, uplo::Symbol) = true
 for (SymHerm, chk, fun) in
     ((:Symmetric, :issymmetric, :transpose), (:Hermitian, :ishermitian, :adjoint))
     @eval begin
@@ -465,7 +469,11 @@ for (SymHerm, chk, fun) in
             dy = project.parent(dx)
             # Here $chk means this is efficient on same-type.
             # If we could mutate dx, then that could speed up action on dx::Matrix.
-            dz = $chk(dy) ? dy : (dy .+ $fun(dy)) ./ 2
+            # Separate returns avoid a type-unstable `dz`, which allocates on Julia 1.10.
+            if $chk(dy) && _same_uplo(dy, project.uplo)
+                return $SymHerm(project.parent(dy), project.uplo)
+            end
+            dz = (dy .+ $fun(dy)) ./ 2
             return $SymHerm(project.parent(dz), project.uplo)
         end
         # This is an example of a subspace which is not a subtype,
diff --git a/test/projection.jl b/test/projection.jl
index 95046147a..d77788b71 100644
--- a/test/projection.jl
+++ b/test/projection.jl
@@ -284,6 +284,27 @@ struct NoSuperType end
         @test pherm(pherm(reshape(1:9, 3, 3))) == pherm(reshape(1:9, 3, 3))
         @test pherm(rand(ComplexF32, 3, 3, 1)) isa Hermitian{ComplexF64}
 
+        @testset "input with other uplo" begin
+            for dx in (Symmetric(rand(3, 3), :L), Hermitian(rand(3, 3), :L))
+                @test @inferred(psymm(dx)) == dx
+                @test psymm(dx).uplo == 'U'
+            end
+            dx = Hermitian(rand(ComplexF64, 3, 3), :U)
+            @test @inferred(pherm(dx)) == dx
+            @test pherm(dx).uplo == 'L'
+            dx = Symmetric(rand(ComplexF64, 3, 3), :L)
+            @test @inferred(ProjectTo(Symmetric(rand(ComplexF64, 3, 3)))(dx)) == dx
+        end
+
+        @testset "plain matrix input that is already symmetric/hermitian" begin
+            dx = [1.0 2.0 3.0; 2.0 4.0 5.0; 3.0 5.0 6.0]
+            @test @inferred(psymm(dx)) == dx
+            @test psymm(dx) isa Symmetric{Float64}
+            dx = [1.0 2.0+im 3.0; 2.0-im 4.0 5.0im; 3.0 -5.0im 6.0]
+            @test @inferred(pherm(dx)) == dx
+            @test pherm(dx) isa Hermitian{ComplexF64}
+        end
+
         pupp = ProjectTo(UpperTriangular(rand(3, 3)))
         @test pupp(reshape(1:9, 3, 3)) == [1.0 4.0 7.0; 0.0 5.0 8.0; 0.0 0.0 9.0]
         @test pupp(pupp(reshape(1:9, 3, 3))) == pupp(reshape(1:9, 3, 3))