Eppstein tunnelling is a non-symmetric relationship between a few pairs of regular maps.
It only yields another regular map when applied to a regular map such as the cube whose vertices form antipodal pairs. (If a regular map has antipodal sets of more than two points, exactly two of which are vertices, such as those of form {6,3}, it isn't good enough: Eppstein tunnelling yields a result which is not regular.)
If such a regular map is described by
The process is described by David Eppstein in the section "Cube triple cover" of the web page The many faces of the Nauru graph.
The process can be described informally as follows. Each pair of antipodal vertices is removed, and replaced by a tunnel. Each edge which had terminated at a vertex instead enters the tunnel and immediately terminates in a newly-created vertex. Within each tunnel a cyclic zizag of edges (forming a Petrie polygon) connects the newly-created edges.