diff --git a/ext/MatrixAlgebraKitEnzymeExt/MatrixAlgebraKitEnzymeExt.jl b/ext/MatrixAlgebraKitEnzymeExt/MatrixAlgebraKitEnzymeExt.jl
index a7014a463..f834881fd 100644
--- a/ext/MatrixAlgebraKitEnzymeExt/MatrixAlgebraKitEnzymeExt.jl
+++ b/ext/MatrixAlgebraKitEnzymeExt/MatrixAlgebraKitEnzymeExt.jl
@@ -8,6 +8,7 @@ using MatrixAlgebraKit: qr_null_pullback!, lq_null_pullback!
 using MatrixAlgebraKit: eig_pullback!, eigh_pullback!, eig_vals_pullback!, eigh_vals_pullback!
 using MatrixAlgebraKit: svd_pullback!, svd_vals_pullback!
 using MatrixAlgebraKit: left_polar_pullback!, right_polar_pullback!
+using MatrixAlgebraKit: left_polar_pushforward!, right_polar_pushforward!
 using Enzyme
 using Enzyme.EnzymeCore
 using Enzyme.EnzymeCore: EnzymeRules
@@ -117,6 +118,36 @@ for (f, pb) in (
     end
 end
 
+for (f, pf) in (
+        (left_polar!, left_polar_pushforward!),
+        (right_polar!, right_polar_pushforward!),
+    )
+    @eval begin
+        function EnzymeRules.forward(
+                config::EnzymeRules.FwdConfigWidth{1},
+                func::Const{typeof($f)},
+                ::Type{RT},
+                A::Annotation,
+                arg::Annotation{TA},
+                alg::Const{<:MatrixAlgebraKit.AbstractAlgorithm},
+            ) where {RT, TA}
+            $f(A.val, arg.val, alg.val)
+            if !isa(A, Const) && !isa(arg, Const)
+                $pf(A.dval, A.val, arg.val, arg.dval)
+            end
+            if EnzymeRules.needs_primal(config) && EnzymeRules.needs_shadow(config)
+                return arg
+            elseif EnzymeRules.needs_primal(config)
+                return arg.val
+            elseif EnzymeRules.needs_shadow(config)
+                return arg.dval
+            else
+                return nothing
+            end
+        end
+    end
+end
+
 for (f, pb) in (
         (qr_null!, qr_null_pullback!),
         (lq_null!, lq_null_pullback!),
diff --git a/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl b/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
index e2ae96c11..28d56ca65 100644
--- a/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
+++ b/ext/MatrixAlgebraKitMooncakeExt/MatrixAlgebraKitMooncakeExt.jl
@@ -9,6 +9,7 @@ using MatrixAlgebraKit: qr_null_pullback!, lq_null_pullback!
 using MatrixAlgebraKit: eig_pullback!, eigh_pullback!, eig_vals_pullback!
 using MatrixAlgebraKit: eig_trunc_pullback!, eigh_trunc_pullback!, eigh_vals_pullback!
 using MatrixAlgebraKit: left_polar_pullback!, right_polar_pullback!
+using MatrixAlgebraKit: left_polar_pushforward!, right_polar_pushforward!
 using MatrixAlgebraKit: svd_pullback!, svd_trunc_pullback!, svd_vals_pullback!
 using MatrixAlgebraKit: TruncatedAlgorithm
 using LinearAlgebra
@@ -16,7 +17,7 @@ using LinearAlgebra
 Mooncake.tangent_type(::Type{<:MatrixAlgebraKit.AbstractAlgorithm}) = Mooncake.NoTangent
 
 # needed for GPU tests because Mooncake can't differentiate through CUDA kernels
-@is_primitive Mooncake.DefaultCtx Mooncake.ReverseMode Tuple{typeof(zero!), AbstractArray}
+@is_primitive Mooncake.DefaultCtx Tuple{typeof(zero!), AbstractArray}
 function Mooncake.rrule!!(::CoDual{typeof(zero!)}, A_dA::CoDual)
     A, dA = arrayify(A_dA)
     Ac = copy(A)
@@ -28,6 +29,12 @@ function Mooncake.rrule!!(::CoDual{typeof(zero!)}, A_dA::CoDual)
     end
     return A_dA, zero_adjoint
 end
+function Mooncake.frule!!(::Dual{typeof(zero!)}, A_dA::Dual)
+    A, dA = arrayify(A_dA)
+    zero!(A)
+    zero!(dA)
+    return A_dA
+end
 
 # two-argument in-place factorizations like LQ, QR, EIG
 for (f!, f, pb, adj) in (
@@ -40,7 +47,6 @@ for (f!, f, pb, adj) in (
         (:left_polar!, :left_polar, :left_polar_pullback!, :left_polar_adjoint),
         (:right_polar!, :right_polar, :right_polar_pullback!, :right_polar_adjoint),
     )
-
     @eval begin
         @is_primitive Mooncake.DefaultCtx Mooncake.ReverseMode Tuple{typeof($f!), Any, Tuple{<:Any, <:Any}, MatrixAlgebraKit.AbstractAlgorithm}
         function Mooncake.rrule!!(::CoDual{typeof($f!)}, A_dA::CoDual, args_dargs::CoDual, alg_dalg::CoDual{<:MatrixAlgebraKit.AbstractAlgorithm})
@@ -104,6 +110,36 @@ for (f!, f, pb, adj) in (
     end
 end
 
+for (f!, f, pf) in (
+        (:left_polar!, :left_polar, :left_polar_pushforward!),
+        (:right_polar!, :right_polar, :right_polar_pushforward!),
+    )
+    @eval begin
+        @is_primitive Mooncake.DefaultCtx Mooncake.ForwardMode Tuple{typeof($f!), Any, Tuple{<:Any, <:Any}, MatrixAlgebraKit.AbstractAlgorithm}
+        function Mooncake.frule!!(::Dual{typeof($f!)}, A_dA::Dual, args_dargs::Dual, alg_dalg::Dual{<:MatrixAlgebraKit.AbstractAlgorithm})
+            A, dA = arrayify(A_dA)
+            args = Mooncake.primal(args_dargs)
+            dargs = Mooncake.tangent(args_dargs)
+            arg1, darg1 = arrayify(args[1], dargs[1])
+            arg2, darg2 = arrayify(args[2], dargs[2])
+            $f!(A, args, Mooncake.primal(alg_dalg))
+            $pf(dA, A, (arg1, arg2), (darg1, darg2))
+            return args_dargs
+        end
+        @is_primitive Mooncake.DefaultCtx Mooncake.ForwardMode Tuple{typeof($f), Any, MatrixAlgebraKit.AbstractAlgorithm}
+        function Mooncake.frule!!(::Dual{typeof($f)}, A_dA::Dual, alg_dalg::Dual{<:MatrixAlgebraKit.AbstractAlgorithm})
+            A, dA = arrayify(A_dA)
+            output = $f(A, Mooncake.primal(alg_dalg))
+            doutput = Mooncake.zero_tangent(output)
+            output_dual = Dual(output, doutput)
+            arg1, darg1 = arrayify(output[1], doutput[1])
+            arg2, darg2 = arrayify(output[2], doutput[2])
+            $pf(dA, A, (arg1, arg2), (darg1, darg2))
+            return output_dual
+        end
+    end
+end
+
 for (f!, f, pb, adj) in (
         (:qr_null!, :qr_null, :qr_null_pullback!, :qr_null_adjoint),
         (:lq_null!, :lq_null, :lq_null_pullback!, :lq_null_adjoint),
diff --git a/src/MatrixAlgebraKit.jl b/src/MatrixAlgebraKit.jl
index 25dc60c1b..37b964b44 100644
--- a/src/MatrixAlgebraKit.jl
+++ b/src/MatrixAlgebraKit.jl
@@ -129,6 +129,8 @@ include("pullbacks/eigh.jl")
 include("pullbacks/svd.jl")
 include("pullbacks/polar.jl")
 
+include("pushforwards/polar.jl")
+
 include("precompile.jl")
 
 end
diff --git a/src/pushforwards/polar.jl b/src/pushforwards/polar.jl
new file mode 100644
index 000000000..6238ffa4c
--- /dev/null
+++ b/src/pushforwards/polar.jl
@@ -0,0 +1,25 @@
+function left_polar_pushforward!(ΔA, A, WP, ΔWP; kwargs...)
+    W, P = WP
+    ΔW, ΔP = ΔWP
+    mul!(ΔP, adjoint(W), ΔA, +1, 0)
+    K̇ = _sylvester(P, P, adjoint(ΔP) - ΔP)
+    mul!(ΔW, ΔA, inv(P), +1, 0)
+    WᴴdAiP = W' * ΔW
+    mul!(ΔW, W, WᴴdAiP, -1, +1)
+    ΔW = mul!(ΔW, W, K̇, +1, +1)
+    ΔP = mul!(ΔP, K̇, P, -1, +1)
+    return (ΔW, ΔP)
+end
+
+function right_polar_pushforward!(ΔA, A, PWᴴ, ΔPWᴴ; kwargs...)
+    P, Wᴴ = PWᴴ
+    ΔP, ΔWᴴ = ΔPWᴴ
+    mul!(ΔP, ΔA, adjoint(Wᴴ), +1, 0)
+    K̇ = _sylvester(P, P, adjoint(ΔP) - ΔP)
+    mul!(ΔWᴴ, inv(P), ΔA, +1, 0)
+    iPdAW = ΔWᴴ * Wᴴ'
+    mul!(ΔWᴴ, iPdAW, Wᴴ, -1, +1)
+    ΔWᴴ = mul!(ΔWᴴ, K̇, Wᴴ, +1, +1)
+    ΔP = mul!(ΔP, P, K̇, -1, +1)
+    return (ΔWᴴ, ΔP)
+end
diff --git a/test/testsuite/enzyme/polar.jl b/test/testsuite/enzyme/polar.jl
index bfc889c24..e342f416d 100644
--- a/test/testsuite/enzyme/polar.jl
+++ b/test/testsuite/enzyme/polar.jl
@@ -14,14 +14,14 @@ end
 """
     test_enzyme_left_polar(T, sz; rng, atol, rtol)
 
-Test the Enzyme reverse-mode AD rule for `left_polar` and its in-place variant. Only runs
-for tall or square matrices (`m >= n`).
+Test the Enzyme forward- and reverse-mode AD rule for `left_polar` and its in-place variant.
+Only runs for tall or square matrices (`m >= n`).
 """
 function test_enzyme_left_polar(
         T, sz;
         rng = Random.default_rng(), atol::Real = 0, rtol::Real = precision(T)
     )
-    return @testset "left_polar reverse: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
+    return @testset "left_polar: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
         A = instantiate_matrix(T, sz)
         m, n = size(A)
         if m >= n
@@ -29,6 +29,9 @@ function test_enzyme_left_polar(
             WP, ΔWP = ad_left_polar_setup(A)
             test_reverse(left_polar, RT, (A, TA), (alg, Const); atol, rtol)
             test_reverse(call_and_zero!, RT, (left_polar!, Const), (A, TA), (alg, Const); atol, rtol)
+            A = instantiate_matrix(T, sz)
+            test_forward(left_polar, RT, (A, TA), (alg, Const); atol, rtol)
+            test_forward(call_and_zero!, RT, (left_polar!, Const), (A, TA), (alg, Const); atol, rtol)
         end
     end
 end
@@ -36,14 +39,14 @@ end
 """
     test_enzyme_right_polar(T, sz; rng, atol, rtol)
 
-Test the Enzyme reverse-mode AD rule for `right_polar` and its in-place variant. Only runs
-for wide or square matrices (`m <= n`).
+Test the Enzyme forward- and reverse-mode AD rule for `right_polar` and its in-place variant.
+Only runs for wide or square matrices (`m <= n`).
 """
 function test_enzyme_right_polar(
         T, sz;
         rng = Random.default_rng(), atol::Real = 0, rtol::Real = precision(T)
     )
-    return @testset "right_polar reverse: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
+    return @testset "right_polar: RT $RT, TA $TA" for RT in (Duplicated,), TA in (Duplicated,)
         A = instantiate_matrix(T, sz)
         m, n = size(A)
         if m <= n
@@ -51,6 +54,9 @@ function test_enzyme_right_polar(
             PWᴴ, ΔPWᴴ = ad_right_polar_setup(A)
             test_reverse(right_polar, RT, (A, TA), (alg, Const); atol, rtol)
             test_reverse(call_and_zero!, RT, (right_polar!, Const), (A, TA), (alg, Const); atol, rtol)
+            A = instantiate_matrix(T, sz)
+            test_forward(right_polar, RT, (A, TA), (alg, Const); atol, rtol)
+            test_forward(call_and_zero!, RT, (right_polar!, Const), (A, TA), (alg, Const); atol, rtol)
         end
     end
 end
diff --git a/test/testsuite/mooncake/polar.jl b/test/testsuite/mooncake/polar.jl
index 161360d62..fee32e71e 100644
--- a/test/testsuite/mooncake/polar.jl
+++ b/test/testsuite/mooncake/polar.jl
@@ -14,8 +14,8 @@ end
 """
     test_mooncake_left_polar(T, sz; rng, atol, rtol)
 
-Test the Mooncake reverse-mode AD rule for `left_polar` and its in-place variant. Only runs
-for tall or square matrices (`m >= n`).
+Test the Mooncake forward- and reverse-mode AD rule for `left_polar` and its in-place variant.
+Only runs for tall or square matrices (`m >= n`).
 """
 function test_mooncake_left_polar(
         T, sz;
@@ -31,11 +31,11 @@ function test_mooncake_left_polar(
 
         Mooncake.TestUtils.test_rule(
             rng, left_polar, A, alg;
-            mode = Mooncake.ReverseMode, output_tangent, atol, rtol
+            output_tangent, atol, rtol
         )
         Mooncake.TestUtils.test_rule(
             rng, call_and_zero!, left_polar!, A, alg;
-            mode = Mooncake.ReverseMode, output_tangent, atol, rtol, is_primitive = false
+            output_tangent, atol, rtol, is_primitive = false
         )
     end
 end
@@ -43,8 +43,8 @@ end
 """
     test_mooncake_right_polar(T, sz; rng, atol, rtol)
 
-Test the Mooncake reverse-mode AD rule for `right_polar` and its in-place variant. Only runs
-for wide or square matrices (`m <= n`).
+Test the Mooncake forward- and reverse-mode AD rule for `right_polar` and its in-place variant.
+Only runs for wide or square matrices (`m <= n`).
 """
 function test_mooncake_right_polar(
         T, sz;
@@ -60,11 +60,11 @@ function test_mooncake_right_polar(
 
         Mooncake.TestUtils.test_rule(
             rng, right_polar, A, alg;
-            mode = Mooncake.ReverseMode, output_tangent, atol, rtol
+            output_tangent, atol, rtol
         )
         Mooncake.TestUtils.test_rule(
             rng, call_and_zero!, right_polar!, A, alg;
-            mode = Mooncake.ReverseMode, output_tangent, atol, rtol, is_primitive = false
+            output_tangent, atol, rtol, is_primitive = false
         )
     end
 end