  Let ${\cal M}$ be a map with the underlying graph $\Gamma$. The automorphism
group $Aut({\cal M})$ induces a natural action on the set of all
vertex-edge-face incident triples, called {\em flags} of ${\cal M}$. The map
${\cal M}$ is said to be a {\em $k$-orbit} map if $Aut({\cal M})$ has $k$
orbits on the set of all flags of ${\cal M}$. It is known that there are seven
different classes of $2$-orbit maps, with only four of them corresponding to
arc-transitive maps, that is maps for which $Aut{\cal M}$ acts arc-transitively
on the underlying graph $\Gamma$. The Petrie dual operator links these four
classes in two pairs, one of which corresponds to the chiral maps and their
Petrie duals.
  In this paper we focus on the other pair of classes of $2$-orbit
arc-transitive maps. We investigate the connection of these maps to consistent
cycles of the underlying graph with special emphasis on such maps of smallest
possible valence, namely $4$. We then give a complete classification of such
maps whose underlying graphs are arc-transitive Rose Window graphs.
