The effects of symmetry and degeneracy on the interactions between spatial modes are studied for parametrically excited surface waves in a rectangular cell. Elementary symmetry considerations are used to make predictions about the phase space structure. However, experimental observations show that in the presence of exact degeneracy (obtained by using a square cell), the phase space dynamics do not show the expected symmetries, and are very complex. On the other hand, when the degeneracy is removed by making the two lateral dimensions slightly different, the observations are consistent with expectations based on symmetry. Exact degeneracy leaves the system very sensitive to small symmetry-breaking perturbations that dramatically affect the dynamics.