  We characterize cutting sequences of infinite geodesics on square-tiled
surfaces by considering interval exchanges on specially chosen intervals on the
surface. These interval exchanges can be thought of as skew products over a
rotation, and we convert cutting sequences to symbolic trajectories of these
interval exchanges to show that special types of combinatorial lifts of
Sturmian sequences completely describe all cutting sequences on a square-tiled
surface. Our results extend the list of families of surfaces where cutting
sequences are understood to a dense subset of the moduli space of all
translation surfaces.
