Is there a commutative operation for which the inverse of the operation is also commutative? For instance addition is commutative, but the inverse, subtraction, is not.
$$
5+2 = 2+5\\
5-2 \neq 2-5
$$
Same for multiplication/division:
$$
5\times4 = 4\times5\\
5/4 \neq 4/5
$$
So is there a group operation $\circ$ with the inverse $\circ^{-1}$ such that
$$
a\circ b = b\circ a\\
a\circ^{-1}b = b\circ^{-1}a
$$

If we define $\circ^{-1}$ for a group, we have a neutral element $1$. Applying commutativity with b = 1 we get
$$1\circ^{-1} a = a \circ^{-1} 1$$
but this simplifies to
$$a^{-1} = a$$
so each element is its own inverse.
