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 $f(0) = 1$.

In a sense, every such bijection is going to look the same. If $a,b \in X$, then we will either have things that look like $f(a) = a$ or $f(a) = b, f(b) = a$ (which I'm going to refer to as a single 'transposition.'
But this gives us infinitely many examples to choose from, even just in $\mathbb{Z}$. You might let $f$ be the identity on every element except, say, $1$ and $5$, such that $f(1) = 5, f(5) = 1$. Or you can have as many transpositions as you'd like.
