Half-regular Cayley maps

A Cayley map M=CM(G,X,P) is an embedding of a Cayley graph C(G,X) in an orientable surface with the property that each left multiplication by an element of G becomes a map automorphism of M: G < Aut(M). A  Cayley map CM(G,X,P) is regular (i.e., its orientation preserving automorphisms act regularly on darts) if and only if a skew-morphism exists that has an orbit X on which it is equal to P. 

With Roman Nedela we asked the question whether the skew-morphisms which do not have this property are good for anything, and arrived at the concept of a half-regular Cayley map.