We present a comprehensive study of the bifurcation routes in a generic model of a two-polarization-mode semiconductor laser when subject to orthogonal optical injection, relevant to a single-transverse-mode vertical-cavity surface-emitting laser. For positive detunings, polarization switching (PS) occurs as the intensity of the noninjected mode gradually decreases and eventually shuts down, in some cases accompanied along the way by the onset and the subsequent vanishing of relaxation oscillations. We identify a Hopf bifurcation on a two-polarization-mode solution, which supports the onset of relaxation oscillation time-periodic dynamics in both injected (x-linearly polarized) and noninjected (y-linearly polarized) modes. For sufficiently large negative detunings, PS is a sharp transition occurring either at the injection-locking point or at a subcritical Hopf bifurcation and is often part of a hysteresis cycle.