  Generalised spin structures, or r-spin structures, on a 2-dimensional
orbifold \Sigma are r-fold fibrewise connected coverings (also called r-th
roots) of its unit tangent bundle ST\Sigma. We investigate such structures on
hyperbolic orbifolds. The conditions on r for such structures to exist are
given. The action of the diffeomorphism group of \Sigma on the set of r-spin
structures is described, and we determine the number of orbits under this
action and their size. These results are then applied to describe the moduli
space of taut contact circles on left-quotients of the 3-dimensional geometry
\tilde{SL}_2.
