Quasiparticles in semiconductors—such as microcavity polaritons—can form condensates in which the steady-state density profile is set by the balance of pumping and decay. By taking account of the polarization degree of freedom for a polariton condensate, and considering the effects of an applied magnetic field, we theoretically discuss the interplay between polarization dynamics, and the spatial structure of the pumped decaying condensate. If spatial structure is neglected, this dynamics has attractors that are linearly polarized condensates (fixed points), and desynchronized solutions (limit cycles), with a range of bistability. Considering spatial fluctuations about the fixed point, the collective spin modes can either be diffusive, linearly dispersing, or gapped. Including spatial structure, interactions between the spin components can influence the dynamics of vortices; produce stable complexes of vortices and rarefaction pulses with both co- and counter-rotating polarizations; and increase the range of possible limit cycles for the polarization dynamics, with different attractors displaying different spatial structures.