We investigate experimentally and theoretically nonlinear dynamics and polarization bistability in a vertical-cavity surface-emitting laser (VCSEL) submitted to an external optical injection with orthogonal polarization, i.e., the injected light from a master laser (ML) is linearly polarized and orthogonal to the polarization direction of the free-running VCSEL (slave laser, SL). Depending on the injection power level and the frequency detuning between ML and SL, our experimental results show that the injected VCSEL may undergo complex dynamics including subharmonic resonances and period-doubling route to chaos that are associated to polarization switching (PS). Using continuation method, we carry out a detailed bifurcation analysis and report on qualitatively different dynamics as we change the injection strength and/or the detuning. We show that PS may be strongly influenced by the presence of those dynamics. Our numerical results reveal period doubling dynamics that emerge from a Hopf bifurcation associated to an elliptical polarized injection locking state. Pure frequency-induced polarization bistability, i.e., when the detuning is changed for fixed injection power, is studied theoretically. Our results show that, depending on the level of the fixed injection strength, frequency induced PS may be achieved with or without hysteresis.