  This paper compares different pseudo-Anosov maps coming from different
Birkhoff sections of a given flow. More precisely, given a hyperbolic surface
and a collection of periodic geodesics on it, we study those Birkhoff sections
for the geodesic flow on the unit bundle to the surface bounded by the
collection. We show that there is a canonical identification of all those
surfaces, and that the first-return maps induced by the flow can all be
expressed as a composition of negative Dehn twists along the same family of
curves: only the order depends on the choice of a particular surface.
