  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 noncrossing
nets of a common polyhedron. Furthermore, monotone reversible hinged
dissections (where all hinges rotate in the same direction when changing from
$A$ to $B$) correspond exactly to noncrossing nets of a common convex
polyhedron. By envelope/parcel magic, it becomes easy to design many hinged
dissections.
