We investigate experimentally and theoretically the polarization switching found in single-transverse-mode vertical-cavity surface-emitting lasers (VCSELs) when subject to parallel optical injection. Our analysis focuses on a recently observed state in which injection locking of the parallel polarization and excitation of the free-running orthogonal polarization of the VCSEL are obtained. A simple nonlinear dependence between the power of both linear polarizations and the frequency detuning is found. Also, the total power emitted by the VCSEL is constant and independent of the injected optical power and the frequency detuning. We check these results experimentally for a variety of frequency detunings and bias currents applied to the device. We report experimental and theoretical stability maps in the injected power-frequency detuning plane for different bias currents, identifying the regions in which the state is observed. A simple analytical expression that describes the map boundary for large and negative frequency detunings is obtained. This provides a simple method to extract the linear dichroism of the device.