We report on the dynamics that accompany polarization switching induced by orthogonal optical injection in a vertical-cavity surface-emitting laser (VCSEL). As the injection strength increases, the VCSEL bifurcates to injection-locked steady states and time-periodic and possibly chaotic dynamics. Of particular interest is a two-mode injection-locking solution, i.e., locking of the two VCSEL polarization modes. A detailed stability analysis unveils new bistability mechanisms in optical injection problems.