If dark matter has spin 0, only two WIMP-nucleon interaction operators can arise as leading operators from the nonrelativistic reduction of renormalizable single-mediator models for dark matter-quark interactions. Based on this crucial observation, we show that about 100 signal events at next generation directional detection experiments can be enough to enable a 2 sigma rejection of the spin 0 dark matter hypothesis in favor of alternative hypotheses where the dark matter particle has spin 1/2 or 1. In this context, directional sensitivity is crucial since anisotropy patterns in the sphere of nuclear recoil directions depend on the spin of the dark matter particle. For comparison, about 100 signal events are expected in a CF4\ detector operating at a pressure of 30 torr with an exposure of approximately 26,000 cubic-meter-detector days for WIMPs of 100 GeV mass and a WIMP-fluorine scattering cross section of 0.25 pb. Comparable exposures require an array of cubic meter time projection chamber detectors.