Itineraries of rigid rotations and diffeomorphisms of the circle

We examine the itinerary of 0∈S1=R/Z under the rotation by α∈R∖Q. The motivating question is: if we are given only the itinerary of 0 relative to I⊂S1, a finite union of closed intervals, can we recover α and I? We prove that the itineraries do determine α and I up to certain equivalences. Then we present elementary methods for finding α and I. Moreover, if g:S1→S1 is a C2, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences.