  Regular maps on linear fractional groups $PSL(2,q)$ and $PGL(2,q$) have been
studied for many years and the theory is well-developed, including generating
sets for the asscoiated groups. This paper studies the properties of
self-duality, self-Petrie-duality and M\"obius regularity in this context,
providing necessary and sufficient conditions for each case. We also address
the special case for regular maps of type (5,5). The final section includes an
enumeration of the $PSL(2,q)$ maps for $q\le81$ and a list of all the
$PSL(2,q)$ maps which have any of these special properties for $q\le49$.
