The two-channel Kondo lattice hosts a rich array of phases, perhaps none more interesting than hastatic order, a channel-symmetry breaking heavy Fermi liquid that has a spinorial order parameter. I will discuss our recent insights into the nature of this unusual phase, including its prevalence in one dimension; the experimental and computational consequences of the spinorial order; how it can be realized in intermetallic materials based on non-Kramers ions like Pr, U and Tm; and how topological defects in the hastatic Kondo insulator can be engineered to host mobile Majorana zero modes or other non-Abelian anyons.
