  We consider a natural local dynamic on the set of all rooted planar maps with
$n$ edges that is in some sense analogous to "edge flip" Markov chains, which
have been considered before on a variety of combinatorial structures
(triangulations of the $n$-gon and quadrangulations of the sphere, among
others). We provide the first polynomial upper bound for the mixing time of
this "edge rotation" chain on planar maps: we show that the spectral gap of the
edge rotation chain is bounded below by an appropriate constant times
$n^{-11/2}$. In doing so, we provide a partially new proof of the fact that the
same bound applies to the spectral gap of edge flips on quadrangulations, which
makes it possible to generalise a recent result of the author and Stauffer to a
chain that relates to edge rotations via Tutte's bijection.
