What are some examples of non-identity bijections $f: X \to X$ such that $f^{-1} = f$ One example I can think of is $f: \mathbb{Z_2} \to \mathbb{Z_2}$ given by $f(1) = 0$ and $f(0) = 1$.

Interesting example. $f(x)=-x, x\in \mathbb{R}$, or $f(x)=\frac{1}{x}, x\in \mathbb{R}\backslash \{0\}$ would come to mind first for me. In general such a function can clearly be defined on any group-like structure with inverses, just by defining a function that takes every element to its inverse, and the identity (if it exists) to itself.
