Polyhedral Characterization of Reversible Hinged Dissections

We prove that two polygons A and B have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between A and B) if and only if A and B are two non-crossing nets of a common polyhedron. Furthermore, monotone hinged dissections (where all hinges rotate in the same direction when changing from A to B) correspond exactly to non-crossing nets of a common convex polyhedron. By envelope/parcel magic, it becomes easy to design many hinged dissections.