We study photonic microwave signal generation obtained when single- and multi-transverse-mode vertical-cavity surface-emitting lasers (VCSELs) are subject to two-frequency orthogonal optical injection. Our calculations show that broadly tunable microwave signals can be obtained in these systems. The response of the multi-transverse-mode VCSEL is enhanced with respect to that obtained with a similar single-transverse-mode VCSEL subject to the same two-frequency orthogonal optical injection. The extra degree of freedom given by the multi-transverse-mode operation of the VCSEL under two-frequency orthogonal optical injection enhances the performance of the photonic microwave generation system, because the higher-order transverse mode is excited with a much larger amplitude than that of the fundamental transverse mode. Periodic oscillations are obtained for a very wide range of frequency detunings between the optical injections and transverse modes. A relative maximum of the microwave signal amplitude is obtained when the frequency of one of the optical injections is very close to the frequency of the orthogonally polarized fundamental mode of the VCSEL. Periodic oscillations are demonstrated for symmetric and asymmetric values of the injection strengths. Wide tuning ranges, extended into the THz band, are obtained in our system. Our results show that the proposed microwave signal generation mechanism is independent of the polarization of the master lasers.