The development of collective long-range order via phase transitions occurs by the spontaneous breaking of fundamental symmetries. Magnetism is the consequence of broken time-reversal symmetry while superfluidity results from broken gauge invariance. The broken symmetry that develops below 17.5 K in the heavy fermion compound URu2Si2 has long eluded such identification. In this talk, I'll discuss how the recent observation of Ising quasiparticles in URu2Si2 indicates a spinor order parameter that breaks double-time reversal symmetry, mixing states of integer and half-integer spin. Such "hastatic order" hybridizes conduction electrons with Ising 5f2 states of the uranium atoms to produce Ising quasiparticles; it accounts for the large entropy of condensation and the magnetic anomaly observed in torque magnetometry. We also have a number of predictions for experiment: hastatic order results in a tiny transverse moment in the conduction sea, a collosal Ising anistropy in the nonlinear susceptibility anomaly and a resonant energy-dependent nematicity in the tunnelling density of states.