  An abstract $n$-polytope $\mathcal{P}$ is a partially-ordered set which
captures important properties of a geometric polytope, for any dimension $n$.
For even $n \ge 2$, the incidences between elements in the middle two layers of
the Hasse diagram of $\mathcal{P}$ give rise to the medial layer graph of
$\mathcal{P}$, denoted by $\mathcal{G} = \mathcal{G}(\mathcal{P})$. If $n=4$,
and $\mathcal{P}$ is both highly symmetric and self-dual of type $\{p,q,p\}$,
then a Cayley graph $\mathcal{C}$ covering $\mathcal{G}$ can be constructed on
a group of polarities of $\mathcal{P}$. In this paper we address some open
questions about the relationship between $\mathcal{G}$ and $\mathcal{C}$ that
were raised in a 2008 paper by Monson and Weiss, and describe some interesting
examples of these graphs. In particular, we give the first known examples of
improperly self-dual chiral polytopes of type $\{3,q,3\}$, which are also among
the very few known examples of highly symmetric self-dual finite polytopes that
do not admit a polarity. Also we show that if $p=3$ then $\mathcal{C}$ cannot
have a higher degree of $s$-arc-transitivity than $\mathcal{G}$, and we present
a family of regular $4$-polytopes of type $\{6,q,6\}$ for which the
vertex-stabilisers in the automorphism group of $\mathcal{C}$ are larger than
those for $\mathcal{G}$.
