diff --git a/src/DifferentiationTest/DifferentiationTest.jl b/src/DifferentiationTest/DifferentiationTest.jl
index 1fdc45393..928811b33 100644
--- a/src/DifferentiationTest/DifferentiationTest.jl
+++ b/src/DifferentiationTest/DifferentiationTest.jl
@@ -16,10 +16,12 @@ using ..DifferentiationInterface
 import ..DifferentiationInterface as DI
 using ..DifferentiationInterface:
     AutoTaped,
+    inner,
     mode,
     mysimilar,
     myzero,
     myzero!!,
+    outer,
     supports_mutation,
     supports_pushforward,
     supports_pullback
diff --git a/src/DifferentiationTest/printing.jl b/src/DifferentiationTest/printing.jl
index cbe65ff0a..db174181d 100644
--- a/src/DifferentiationTest/printing.jl
+++ b/src/DifferentiationTest/printing.jl
@@ -34,3 +34,7 @@ function backend_string(backend::AbstractADType)
         error("Unknown mode")
     end
 end
+
+function backend_string(backend::SecondOrder)
+    return "$(backend_string(outer(backend))) / $(backend_string(inner(backend)))"
+end
diff --git a/src/hvp.jl b/src/hvp.jl
index fafe2ad94..213cc7056 100644
--- a/src/hvp.jl
+++ b/src/hvp.jl
@@ -13,12 +13,6 @@ By order of preference:
 """
     hvp(f, backend, x, v, [extras]) -> p
 """
-function hvp(
-    f::F, backend::AbstractADType, x::Number, v, extras=prepare_hvp(f, backend, x)
-) where {F}
-    return v * second_derivative(f, backend, x, extras)
-end
-
 function hvp(
     f::F, backend::AbstractADType, x, v, extras=prepare_hvp(f, backend, x)
 ) where {F}
@@ -27,7 +21,13 @@ function hvp(
     return hvp(f, new_backend, x, v, new_extras)
 end
 
-function hvp(f::F, backend::SecondOrder, x, v, extras=prepare_hvp(backend, f, x)) where {F}
+function hvp(
+    f::F, backend::SecondOrder, x::Number, v::Number, extras=prepare_hvp(f, backend, x)
+) where {F}
+    return v * second_derivative(f, backend, x, extras)
+end
+
+function hvp(f::F, backend::SecondOrder, x, v, extras=prepare_hvp(f, backend, x)) where {F}
     return hvp_aux(f, backend, x, v, extras, hvp_mode(backend))
 end
 
diff --git a/test/runtests.jl b/test/runtests.jl
index 2f4d006a4..41bf8588f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -98,4 +98,8 @@ using Zygote: Zygote
             @time include("zygote.jl")
         end
     end
+
+    @testset verbose = true "Second order" begin
+        include("second_order.jl")
+    end
 end;
diff --git a/test/second_order.jl b/test/second_order.jl
new file mode 100644
index 000000000..232f009f1
--- /dev/null
+++ b/test/second_order.jl
@@ -0,0 +1,31 @@
+using ADTypes
+using DifferentiationInterface
+using DifferentiationInterface.DifferentiationTest
+using DifferentiationInterface.DifferentiationTest: backend_string
+
+using FiniteDiff: FiniteDiff
+using ForwardDiff: ForwardDiff
+using Enzyme: Enzyme
+using Zygote: Zygote
+
+using JET: JET
+using Test
+
+SECOND_ORDER_BACKENDS = Dict(
+    "forward/forward" => [
+        SecondOrder(AutoEnzyme(Enzyme.Forward), AutoForwardDiff()),
+        SecondOrder(AutoForwardDiff(), AutoEnzyme(Enzyme.Forward)),
+    ],
+    "forward/reverse" => [SecondOrder(AutoForwardDiff(), AutoZygote())],
+    "reverse/forward" => [],
+)
+
+@testset verbose = true "Cross backends" begin
+    @testset verbose = true "$second_order_mode" for (second_order_mode, backends) in
+                                                     pairs(SECOND_ORDER_BACKENDS)
+        @info "Testing $second_order_mode..."
+        @time @testset "$(backend_string(backend))" for backend in backends
+            test_operators(backend; first_order=false, type_stability=false)
+        end
+    end
+end;
diff --git a/test/zero.jl b/test/zero.jl
index 39881e341..f05f5a814 100644
--- a/test/zero.jl
+++ b/test/zero.jl
@@ -34,9 +34,8 @@ test_operators(
 
 test_operators(
     [AutoZeroForward(), AutoZeroReverse()];
-    allocating=false,
     correctness=false,
-    type_stability=true,
+    type_stability=false,
     allocations=true,
 );