Network architecture can have a large impact on the dynamics of coupled phase oscillators. It will be shown that architecture can force relations between average frequencies of the different oscillators in such networks. The main tool in this analysis is coupled cell theory which provides precise relations between network architecture and the flow-invariance of certain polydiagonal subspaces. Architecture also imposes restrictions on the spatiotemporal symmetries of periodic solutions that a network of phase oscillators can support. These obstructions are related, but distinct from those observed in equivariant differential equations.