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 WTe 2 , 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 WTe 2 /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. © 2017 Macmillan Publishers Limited.