  We call "flippable tilings" of a constant curvature surface a tiling by
"black" and "white" faces, so that each edge is adjacent to two black and two
white faces (one of each on each side), the black face is forward on the right
side and backward on the left side, and it is possible to "flip" the tiling by
pushing all black faces forward on the left side and backward on the right
side. Among those tilings we distinguish the "symmetric" ones, for which the
metric on the surface does not change under the flip. We provide some existence
statements, and explain how to parameterize the space of those tilings (with a
fixed number of black faces) in different ways. For instance one can glue the
white faces only, and obtain a metric with cone singularities which, in the
hyperbolic and spherical case, uniquely determines a symmetric tiling. The
proofs are based on the geometry of polyhedral surfaces in 3-dimensional spaces
modeled either on the sphere or on the anti-de Sitter space.
