We report on the observation of magnetic-field-induced photocurrent in HgTe/HgCdTe quantum wells of different widths. Both the intrasubband and interband absorption of infrared/terahertz radiation in the heterostructures is shown to cause a dc electric current in the presence of an in-plane magnetic field. The photocurrent behavior upon variation in the radiation polarization, magnetic-field strength, and temperature is studied. At a moderate magnetic field the current exhibits a linear field dependence. At high magnetic fields, however, it becomes nonlinear and is dominated by a cubic in magnetic-field contribution. The latter effect is observed in quantum wells with the inverted band structure only. The experimental results are analyzed in terms of the phenomenological theory and microscopic models of magnetogyrotropic photogalvanic effect based on asymmetry of optical transitions and/or asymmetric relaxation of carriers in the momentum space. The effect is shown to be related to the gyrotropic properties of the structures. The developed theory of magnetogyrotropic photocurrent describes well all experimental results. It is shown that both intrasubband and interband optical transitions may lead to spin-related as well as to spin-independent magnetic-field-induced photocurrents.