Suppose you have a $(2n)$-gon for some $n \in \mathbb{N} : n > 1$. Then the rotation $r^{\frac{n}{2}}$ where $n$ is the number of vertices imposed on the $(2n)$-gon is the same as an involution.
By same, I don't really know what I mean. It gives the same result, but the function isn't the same. Is it possible to define it?
Hint: Write down the matrix of $r$ as $$ \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} $$ where $\theta = 2\pi/2n = \pi/n$. Now what happens when you raise to the power $n/2$?