  In this work for the first time we enumerate unlabelled maps on orientable
genus $g$ surfaces with respect to all homeomorphisms, including both
orientation-preserving and orientation-reversing. We show that in the latter
case as an intermediate step one has to enumerate rooted maps of a special kind
(quotient maps) on orientable and non-orientable surfaces possibly having a
boundary and a certain number of branch points. In this work we develop a
special technique for enumerating such maps.
