The locations of periodic points for Stokes flow in a rectangular cavity is studied. Discontinuous and sinusoidal corotating periodic motions of the top and bottom walls are considered. An effective algorithm for one-dimensional search based on the use of symmetry conditions of the flow is proposed for both cases. It permits one to find and classify all periodic points of low-order periodicity. Specific features of the bifurcation diagrams are noticed. Two typical examples of flow with the dye blob situated around an elliptic or a hyperbolic periodic point show a striking difference in mixing properties.