Regular maps consist of triples $(G,a,b)$ where $G$ is a finite group and $(a,b)$ is a pair of generators of $G$ such that the product $ab$ is an involution. A regular map ${\cal M}=(G,a,b)$ is reflexible, or chiral, according as $\cal M$ is isomorphic, or not, to its mirror image $\overline{{\cal M}}=(G,a^{-1},b^{-1})$.

A few years ago there was a widespread conviction that the ratio of the number of regular reflexible maps up to size $n$ (order of the group $G$) to the number of regular chiral maps up to size $n$, would be asymptotically zero. A calculation carried out on the Suzuki groups by Dimitri and Hubard in 2012 seems to support this idea. However the ratio can be (interestingly) understood in several restrictive ways, for example up to genera $n$, or among regular maps of a fixed number of faces, or a fixed family of groups (as Dimitri and Huberd did), or a fixed family of underlying graphs, etc.

 
In this talk we show how in a restricted family of regular oriented maps with fixed number of ``faces" (orbits of $a$) one can have different, and sometimes surprising, results.