Active particles, such as motor proteins and motile micro-organisms, convert chemical energy from a reservoir to do mechanical work in their environment. We are interested in cases where this energy conversion process involves a periodic duty cycle, such as a cyclic swimming stroke or a periodic stepping of a motor, and can be mapped onto a phase variable. We study the hydrodynamics of a suspension of such particles. For an orientationally aligned suspension of active particles, a stability analysis predicts that both the phase-ordered and disordered states are unstable. Nonlinear simulations of such aligned suspensions reveal the formation of chimera states. The phase dynamics can further trigger alignment instabilities that are unique to these systems and are absent in classical active fluids with time-periodic force dipoles. We find that in channels, a combination of the alignment and phase instability allows these active particles to self-organize and generate unidirectional pumping.