What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it.

Additive inverse
Multiplicative inverse
Fourier transform
Complex conjugation
Any group built up from $\mathbb{Z}_2$, applying (one of) the $\mathbb{Z}_2$ parts' operation.

"Idempotent" came to mind, but that's wrong. It means $f(f(x)) = f(x)$, not $f(f(x))=x$.

What is the word for this "flip-flop" property?

Solution 1:

Involution is the most common name. They are so fundamental that an entire book has been written on them, the Book of Involutions. I often emphasize their essential role both here and various other places. One should always strive to bring to the fore the innate symmetries in problems, and involutions are one of the simplest examples.

Note: you could have found the answer simply by Googling "self inverse function". The first match is the "self inverse" section of the Wikipedia page on inverse functions, which states "such a function is called an involution".