What other involutions are there besides xor?
Roll your own.

Take an arbitrary bijection $f$ from the integers to integers. This is a permutation of the integers.

Then, take the conjugate of XOR ($\oplus$):
$$
    g(x,y) = f^{-1}(f(x) \oplus f(y))
$$
We then have
$$
g(x,x) = f^{-1}(0)
$$
which now acts as the "new zero" (neutral element)
$$
g(x,f^{-1}(0)) = x
$$
We also have that $g$ is associative and commutative
$$
g(x,y) = g(y,z) \qquad
g(g(x,y),z) = g(x,g(y,z))
$$
So, we still have an abelian group where $x^{-1}=x$. The trick on arrays mentioned by the OP still works.