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Recent discoveries regarding current-induced spin–orbit torques produced by heavy-metal/ferromagnet and topological-insulator/ferromagnet bilayers provide the potential for dramatically improved efficiency in the manipulation of magnetic devices. However, in experiments performed to date, spin–orbit torques have an important limitation—the component of torque that can compensate magnetic damping is required by symmetry to lie within the device plane. This means that spin–orbit torques can drive the most current-efficient type of magnetic reversal (antidamping switching) only for magnetic devices with in-plane anisotropy, not the devices with perpendicular magnetic anisotropy that are needed for high-density applications. Here we show experimentally that this state of affairs is not fundamental, but rather one can change the allowed symmetries of spin–orbit torques in spin-source/ferromagnet bilayer devices by using a spin-source material with low crystalline symmetry. We use WTe2, a transition-metal dichalcogenide whose surface crystal structure has only one mirror plane and no two-fold rotational invariance. Consistent with these symmetries, we generate an out-of-plane antidamping torque when current is applied along a low-symmetry axis of WTe2/Permalloy bilayers, but not when current is applied along a high-symmetry axis. Controlling spin–orbit torques by crystal symmetries in multilayer samples provides a new strategy for optimizing future magnetic technologies.