  We derive self-reciprocity properties for a number of polyomino generating
functions, including several families of column-convex polygons, three-choice
polygons and staircase polygons with a staircase hole. In so doing, we
establish a connection between the reciprocity results known to
combinatorialists and the inversion relations used by physicists to solve
models in statistical mechanics. For several classes of convex polygons, the
inversion (reciprocity) relation, augmented by certain symmetry and analyticity
properties, completely determines the anisotropic perimeter generating
function.
