diff --git a/stdlib/LinearAlgebra/src/structuredbroadcast.jl b/stdlib/LinearAlgebra/src/structuredbroadcast.jl
index fcd5c68d48abe..68e0a163af664 100644
--- a/stdlib/LinearAlgebra/src/structuredbroadcast.jl
+++ b/stdlib/LinearAlgebra/src/structuredbroadcast.jl
@@ -102,6 +102,7 @@ function Base.similar(bc::Broadcasted{StructuredMatrixStyle{T}}, ::Type{ElType})
 end
 
 function copyto!(dest::Diagonal, bc::Broadcasted{<:StructuredMatrixStyle})
+    !isstructurepreserving(bc) && !fzeropreserving(bc) && copyto!(dest, convert(Broadcasted{Nothing}, bc))
     axs = axes(dest)
     axes(bc) == axs || Broadcast.throwdm(axes(bc), axs)
     for i in axs[1]
@@ -111,6 +112,7 @@ function copyto!(dest::Diagonal, bc::Broadcasted{<:StructuredMatrixStyle})
 end
 
 function copyto!(dest::Bidiagonal, bc::Broadcasted{<:StructuredMatrixStyle})
+    !isstructurepreserving(bc) && !fzeropreserving(bc) && copyto!(dest, convert(Broadcasted{Nothing}, bc))
     axs = axes(dest)
     axes(bc) == axs || Broadcast.throwdm(axes(bc), axs)
     for i in axs[1]
@@ -129,18 +131,22 @@ function copyto!(dest::Bidiagonal, bc::Broadcasted{<:StructuredMatrixStyle})
 end
 
 function copyto!(dest::SymTridiagonal, bc::Broadcasted{<:StructuredMatrixStyle})
+    !isstructurepreserving(bc) && !fzeropreserving(bc) && copyto!(dest, convert(Broadcasted{Nothing}, bc))
     axs = axes(dest)
     axes(bc) == axs || Broadcast.throwdm(axes(bc), axs)
     for i in axs[1]
         dest.dv[i] = Broadcast._broadcast_getindex(bc, CartesianIndex(i, i))
     end
     for i = 1:size(dest, 1)-1
-        dest.ev[i] = Broadcast._broadcast_getindex(bc, CartesianIndex(i, i+1))
+        v = Broadcast._broadcast_getindex(bc, CartesianIndex(i, i+1))
+        v == Broadcast._broadcast_getindex(bc, CartesianIndex(i+1, i)) || throw(ArgumentError("broadcasted assignment breaks symmetry between locations ($i, $(i+1)) and ($(i+1), $i)"))
+        dest.ev[i] = v
     end
     return dest
 end
 
 function copyto!(dest::Tridiagonal, bc::Broadcasted{<:StructuredMatrixStyle})
+    !isstructurepreserving(bc) && !fzeropreserving(bc) && copyto!(dest, convert(Broadcasted{Nothing}, bc))
     axs = axes(dest)
     axes(bc) == axs || Broadcast.throwdm(axes(bc), axs)
     for i in axs[1]
@@ -154,6 +160,7 @@ function copyto!(dest::Tridiagonal, bc::Broadcasted{<:StructuredMatrixStyle})
 end
 
 function copyto!(dest::LowerTriangular, bc::Broadcasted{<:StructuredMatrixStyle})
+    !isstructurepreserving(bc) && !fzeropreserving(bc) && copyto!(dest, convert(Broadcasted{Nothing}, bc))
     axs = axes(dest)
     axes(bc) == axs || Broadcast.throwdm(axes(bc), axs)
     for j in axs[2]
@@ -165,6 +172,7 @@ function copyto!(dest::LowerTriangular, bc::Broadcasted{<:StructuredMatrixStyle}
 end
 
 function copyto!(dest::UpperTriangular, bc::Broadcasted{<:StructuredMatrixStyle})
+    !isstructurepreserving(bc) && !fzeropreserving(bc) && copyto!(dest, convert(Broadcasted{Nothing}, bc))
     axs = axes(dest)
     axes(bc) == axs || Broadcast.throwdm(axes(bc), axs)
     for j in axs[2]
diff --git a/stdlib/LinearAlgebra/test/structuredbroadcast.jl b/stdlib/LinearAlgebra/test/structuredbroadcast.jl
index 2999da87094bb..c3aa70e2c964e 100644
--- a/stdlib/LinearAlgebra/test/structuredbroadcast.jl
+++ b/stdlib/LinearAlgebra/test/structuredbroadcast.jl
@@ -51,14 +51,37 @@ end
     A = rand(N, N)
     sA = A + copy(A')
     D = Diagonal(rand(N))
-    B = Bidiagonal(rand(N), rand(N - 1), :U)
+    Bu = Bidiagonal(rand(N), rand(N - 1), :U)
+    Bl = Bidiagonal(rand(N), rand(N - 1), :L)
     T = Tridiagonal(rand(N - 1), rand(N), rand(N - 1))
+    ◣ = LowerTriangular(rand(N,N))
+    ◥ = UpperTriangular(rand(N,N))
+
     @test broadcast!(sin, copy(D), D) == Diagonal(sin.(D))
-    @test broadcast!(sin, copy(B), B) == Bidiagonal(sin.(B), :U)
+    @test broadcast!(sin, copy(Bu), Bu) == Bidiagonal(sin.(Bu), :U)
+    @test broadcast!(sin, copy(Bl), Bl) == Bidiagonal(sin.(Bl), :L)
     @test broadcast!(sin, copy(T), T) == Tridiagonal(sin.(T))
+    @test broadcast!(sin, copy(◣), ◣) == LowerTriangular(sin.(◣))
+    @test broadcast!(sin, copy(◥), ◥) == UpperTriangular(sin.(◥))
     @test broadcast!(*, copy(D), D, A) == Diagonal(broadcast(*, D, A))
-    @test broadcast!(*, copy(B), B, A) == Bidiagonal(broadcast(*, B, A), :U)
+    @test broadcast!(*, copy(Bu), Bu, A) == Bidiagonal(broadcast(*, Bu, A), :U)
+    @test broadcast!(*, copy(Bl), Bl, A) == Bidiagonal(broadcast(*, Bl, A), :L)
     @test broadcast!(*, copy(T), T, A) == Tridiagonal(broadcast(*, T, A))
+    @test broadcast!(*, copy(◣), ◣, A) == LowerTriangular(broadcast(*, ◣, A))
+    @test broadcast!(*, copy(◥), ◥, A) == UpperTriangular(broadcast(*, ◥, A))
+
+    @test_throws ArgumentError broadcast!(cos, copy(D), D) == Diagonal(sin.(D))
+    @test_throws ArgumentError broadcast!(cos, copy(Bu), Bu) == Bidiagonal(sin.(Bu), :U)
+    @test_throws ArgumentError broadcast!(cos, copy(Bl), Bl) == Bidiagonal(sin.(Bl), :L)
+    @test_throws ArgumentError broadcast!(cos, copy(T), T) == Tridiagonal(sin.(T))
+    @test_throws ArgumentError broadcast!(cos, copy(◣), ◣) == LowerTriangular(sin.(◣))
+    @test_throws ArgumentError broadcast!(cos, copy(◥), ◥) == UpperTriangular(sin.(◥))
+    @test_throws ArgumentError broadcast!(+, copy(D), D, A) == Diagonal(broadcast(*, D, A))
+    @test_throws ArgumentError broadcast!(+, copy(Bu), Bu, A) == Bidiagonal(broadcast(*, Bu, A), :U)
+    @test_throws ArgumentError broadcast!(+, copy(Bl), Bl, A) == Bidiagonal(broadcast(*, Bl, A), :L)
+    @test_throws ArgumentError broadcast!(+, copy(T), T, A) == Tridiagonal(broadcast(*, T, A))
+    @test_throws ArgumentError broadcast!(+, copy(◣), ◣, A) == LowerTriangular(broadcast(*, ◣, A))
+    @test_throws ArgumentError broadcast!(+, copy(◥), ◥, A) == UpperTriangular(broadcast(*, ◥, A))
 end
 
 @testset "map[!] over combinations of structured matrices" begin