Commutativity

      {(⍵)(⍉⍵)} ∘.<⍨ ⍳4
┌───────┬───────┐
│0 1 1 1│0 0 0 0│
│0 0 1 1│1 0 0 0│
│0 0 0 1│1 1 0 0│
│0 0 0 0│1 1 1 0│
└───────┴───────┘
      {(⍵)(⍉⍵)} ∘.=⍨ ⍳4
┌───────┬───────┐
│1 0 0 0│1 0 0 0│
│0 1 0 0│0 1 0 0│
│0 0 1 0│0 0 1 0│
│0 0 0 1│0 0 0 1│
└───────┴───────┘
      {⍵≡⍉⍵} ∘.=⍨ ⍳4
1

f is commutative on the list x if and only if ∘.f⍨ x is symmetric ({⍵≡⍉⍵}).