Square-wave switching of the intensities of the orthogonal linearly polarized components of the output of a vertical-cavity surface-emitting laser (VCSEL) found previously in experiments with polarization-changing optical feedback, is not found in rate equation models incorporating only birefringence and gain anisotropy, but is found in the model for VCSELs developed by San Miguel, Feng, and Moloney [M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1729 (1995)]. The dynamics is sensitive to both the feedback strength and the relaxation rate of the magnetization in the quantum well sublevels. © 1999 American Institute of Physics.