Mad weaving on non-orientable surfaces

Since the meadow diagonal of each quadrilateral of anyam gila weaving exactly overlies the meadow diagonal of the quadrilateral on the reverse face, and since every triangle is of the same polarity, it does not matter how we are oriented when we come to a quadrilateral with a marked meadow diagonal, we will label the front and back quadrilaterals the same way coming from either orientation. Therefore there is no problem in mad weaving non-orientable surfaces.

A suitable way to generate surfaces divided into quadrilaterals with marked diagonals consistent with the needs of anyam gila is to start with a bipartite map of the surface (a properly embedded graph with vertices colored black and white, such that no edge connects vertices of the same color,) place a pink vertex in the center of each face and connect new edges between the pink vertex and the black and white vertices surrounding it (barycentric subdivision). The meadow edges are now the pink-white (alternately the pink-black) edges. Once a handedness for the weaving is chosen, all is determined.