Grid square dances share features of both American square dances and contra dances. A number of four-couple ``squares'' are initially arranged in a grid on a large dance floor. Each dance is short and performed multiple times; an iteration of the dance ends with some couples ``progressing'' into a neighboring square. Grid squares are relatively new dances, and there is not an established manner of progressing from square to square. Rather, choreographers such as Bob Isaacs, Rick Mohr, Kathy Anderson, and Carol Ormand have experimented with different patterns of movement within and between squares. We will discuss the aesthetics of grid square progressions and use techniques from the study of discrete-time dynamical systems to analyze particular dances.