What is the name two mutually idempotent functions?
These are known as inverse functions. For example, the mappings $x \mapsto 2x$ and $x \mapsto x/2$ are inverses of each other, as are $\exp$ and $\log$ (restricted to positive numbers), or $\sin$ (restricted to the interval $[-\pi/2,\pi/2]$) and $\arcsin$.