  A number which is either the square of an integer or two times the square of
an integer is called squarish. There are two main results in the literature on
graphs whose number of perfect matchings is squarish: one due to Jockusch (for
planar graphs invariant under rotation by 90 degrees) and the other due to the
second author (concerning planar graphs with two perpendicular symmetry axes).
We present in this paper a new such class, consisting of certain planar graphs
that are only required to have one symmetry axis. Our proof relies on a natural
bijection between the set of perfect matchings of two closely related (but not
isomorphic!) families of graphs, which is interesting in its own right. The
rephrasing of this bijection in terms of spanning trees turns out to be the
most natural way to present this result.
  The basic move in the construction of the above bijection (which we call
gliding) can also be used to extend Temperley's classical bijection between
spanning trees of a planar graph and perfect matchings of a closely related
graph. We present this, and as an application we answer an open question posed
by Corteel, Huang and Krattenthaler.
  We also discuss another dimer bijection (used in the proof of the second
author's result mentioned above), and deduce from a refinement of it new
results for spanning trees. These include a finitary version of an independence
result for the uniform spanning tree on $\Z^2$ due to Johansson, a counterpart
of it, and a bijective proof of an independence result on edge inclusions in
the uniform spanning tree on $\Z^2$ due to Lyons.
