diff --git a/src/lib/array.jl b/src/lib/array.jl
index 1c6e09916..b7c6d45ee 100644
--- a/src/lib/array.jl
+++ b/src/lib/array.jl
@@ -2,6 +2,8 @@ using Random, FillArrays, AbstractFFTs
 using FillArrays: AbstractFill, getindex_value
 using Base.Broadcast: broadcasted, broadcast_shape
 using Distributed: pmap, AbstractWorkerPool
+using LinearAlgebra: Diagonal, Hermitian, LowerTriangular, UpperTriangular
+using LinearAlgebra: UnitLowerTriangular, UnitUpperTriangular
 
 @adjoint Array(xs::AbstractArray) = Array(xs), ȳ -> (ȳ,)
 @adjoint Array(xs::Array) = Array(xs), ȳ -> (ȳ,)
@@ -165,10 +167,21 @@ end
 # This is also used by comprehensions, which do guarantee iteration order.
 # Not done for pmap, presumably because all is lost if you are relying on its order.
 _tryreverse(m, backs, Δ) = backs, Δ
-_tryreverse(m::typeof(map), backs, Δ) = reverse(backs), reverse(Δ)
+_tryreverse(m::typeof(map), backs, Δ) = _reverse(backs), _reverse(Δ)
 
 _tryreverse(m, x) = x
-_tryreverse(m::typeof(map), x) = reverse(x)
+_tryreverse(m::typeof(map), x) = _reverse(x)
+
+# Fallback
+_reverse(x) = reverse(x)
+
+# Known cases in the standard library on which `reverse` errors (issue #1393)
+_reverse(x::LowerTriangular) = UpperTriangular(_reverse(parent(x)))
+_reverse(x::UpperTriangular) = LowerTriangular(_reverse(parent(x)))
+_reverse(x::UnitLowerTriangular) = UnitUpperTriangular(_reverse(parent(x)))
+_reverse(x::UnitUpperTriangular) = UnitLowerTriangular(_reverse(parent(x)))
+_reverse(x::Hermitian) = Hermitian(_reverse(x.data), x.uplo == 'U' ? :L : :U)
+_reverse(x::Symmetric) = Symmetric(_reverse(x.data), x.uplo == 'U' ? :L : :U)
 
 # With mismatched lengths, map stops early. With mismatched shapes, it makes a vector.
 # So we keep axes(x) to restore gradient dx to its full length & correct shape.
diff --git a/test/lib/array.jl b/test/lib/array.jl
index d02e9f9d3..889301c1e 100644
--- a/test/lib/array.jl
+++ b/test/lib/array.jl
@@ -1,5 +1,7 @@
 using ChainRulesTestUtils
-using Zygote: ZygoteRuleConfig, _pullback
+using LinearAlgebra: Diagonal, Hermitian, LowerTriangular, UpperTriangular
+using LinearAlgebra: UnitLowerTriangular, UnitUpperTriangular
+using Zygote: ZygoteRuleConfig, _pullback, _reverse
 
 # issue 897
 
@@ -65,3 +67,32 @@ end
     end
     @test gradient(f_comprehension, w)[1] == ones(5)
 end
+
+@testset "_reverse" begin
+    m = [1 2 3; 4 5 6; 7 8 9]
+    @testset "$wrapper" for wrapper in [
+        Hermitian, Symmetric, LowerTriangular, UpperTriangular,
+        UnitLowerTriangular, UnitUpperTriangular,
+    ]
+        M = wrapper(m)
+        @test collect(_reverse(M)) == _reverse(collect(M))
+    end
+end
+
+@testset "rrule for `map`" begin
+    @testset "MWE from #1393" begin
+        # https://github.com/FluxML/Zygote.jl/issues/1393#issuecomment-1468496804
+        struct Foo1393 x::Float64 end
+        (f::Foo1393)(x) = f.x * x
+        x = randn(5, 5)
+        out, pb = Zygote.pullback(x -> map(Foo1393(5.0), x), x)
+        @testset "$wrapper" for wrapper in [
+            Hermitian, Symmetric, LowerTriangular, UpperTriangular,
+            UnitLowerTriangular, UnitUpperTriangular,
+        ]
+            m = wrapper(rand(5, 5))
+            res = only(pb(m))
+            @test res == 5m
+        end
+    end
+end