Write $f\in S(X)$ as a composition of $g,h\in S(X)$ such that $g$ and $h$ are of order $2$.

Question

I'm working on a group theory exercise in which I have to show that any function in $S(X)$, the group of all bijective functions mapping $X$ to itself, can be written as the composition of two functions of order $2$, that is, functions being their own inverse. Here is $X$ is assumed to be an infinite set. In the first part I managed to show this is true for the group of congruences of $\mathbb{R}^2$ mapping the origin onto itself. However, I have no idea what to do here, since I dit not manage to find any decomposition that seems to work. Any help/hint is much appreciated!