Is there a specific term for a function $f:X\to X$ that obeys the law $f^3(x)=f(x)$? It's not necessarily an involution, but it is an involution when its domain is restricted to its image.
A simple example is the function $f:\{1,2,3\}\to\{1,2,3\}$ defined by $$ 1\mapsto2, 2\mapsto3,3\mapsto2 $$