Symmetries of the Three Gap Theorem

The Three Gap Theorem states that for any $\alpha \in \mathbb{R}$ and $N\in \mathbb{N}$, the fractional parts of $\{0\alpha ,1\alpha ,\dots ,(N-1)\alpha \}$ partition the unit circle into gaps of at most three distinct lengths. We prove a result about symmetries in the order with which the sizes of gaps appear on the circle.