  Baxter permutations are known to be in bijection with a wide number of
combinatorial objects. Previously, it was shown that each of these objects had
a natural involution which was carried equivariantly by the known bijections,
and the number of objects fixed under involution was given by Stembridge's
$q=-1$ phenomenon. In this paper, we consider the order 4 action of a
quarter-turn rotation of a Baxter permutation matrix, refining the half-turn
rotation previously studied. Using the method of generating trees, we show that
the number of Baxter permutations fixed under quarter-turn rotation has a very
nice enumeration, which suggests the existence of a combinatorial bijection.
