We report on the observation of linear and circular magnetogyrotropic photogalvanic effects in InSb/AlInSb quantum well structures. We show that intraband (Drude-like) absorption of terahertz radiation in the heterostructures causes a dc electric current in the presence of an in-plane magnetic field. The photocurrent behavior upon variation of the magnetic field strength, temperature and wavelength is studied. We show that at moderate magnetic fields the photocurrent exhibits a typical linear field dependence. At high magnetic fields, however, it becomes nonlinear and inverses its sign. The experimental results are analyzed in terms of the microscopic models based on asymmetric relaxation of carriers in the momentum space. We demonstrate that the observed nonlinearity of the photocurrent is caused by the large Zeeman spin splitting in InSb/AlInSb structures and an interplay of the spin-related and spin-independent roots of the magnetogyrotropic photogalvanic effect.