Itineraries of rigid rotations and diffeomorphisms of the circle

We examine the itinerary of $0\in S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ under the rotation by $\alpha \in \mathbb{R}\text{bs}\text{Q}$. The motivating question is: if we are given only the itinerary of 0 relative to $I\subset S^1$, a finite union of closed intervals, can we recover $\alpha$ and $I$? We prove that the itineraries do determine $\alpha$ and $I$ up to certain equivalences. Then we present elementary methods for finding $\alpha$ and $I$. Moreover, if $g:S^{1}\to S^{1}$isa$C^{2}$, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences.