In frustrated quantum magnets, long-range magnetic order fails to develop despite a large exchange coupling between the spins. In contrast to the magnons in conventional magnets, their spin excitations are poorly understood. Here, we show that the thermal Hall conductivity kappa(xy) provides a powerful probe of spin excitations in the "quantum spin ice" pyrochlore Tb2Ti2O7. The thermal Hall response is large, even though the material is transparent. The Hall response arises from spin excitations with specific characteristics that distinguish them from magnons. At low temperature (<1 kelvin), the thermal conductivity resembles that of a dirty metal. Using the Hall angle, we construct a phase diagram showing how the excitations are suppressed by a magnetic field.