Computational fluid dynamics has been used to study inviscid shock interactions on double-wedge geometries with the purpose of understanding the fundamental gas dynamics of these interactions. The parameter space of the interactions has been explored and the different types of interactions that occur have been identified. Although the interactions are produced by a different geometry, all but one of them may be identified as an Edney Type I, IV, V, or VI interaction. The previously unidentified interaction occurs because of the geometrical constraints imposed by the double wedge. The physical mechanisms for transition have been studied, and the transition criteria have been identified. An important result is that there are two different regimes of the parameter space in which the state of the flow downstream of the interaction point is fundamentally different. At high Mach numbers this flow is characterized by an underexpanded jet which impinges on the wedge and produces large-amplitude surface pressure variations. At low Mach numbers, the jet becomes a shear layer which no longer impinges on the wedge surface.