  Exploiting the relationship between 4-dimensional toric and semitoric
integrable systems with Delzant and semitoric polygons, respectively, we
develop techniques to compute certain equivariant packing densities and
equivariant capacities of these systems by working exclusively with the
polygons. This expands on results of Pelayo and Pelayo-Schmidt. We compute the
densities of several important examples and we also use our techniques to solve
the equivariant semitoric perfect packing problem, i.e., we list all semitoric
polygons for which the associated semitoric system admits an equivariant
packing which fills all but a set of measure zero of the manifold. This paper
also serves as a concise and accessible introduction to Delzant and semitoric
polygons in dimension four.
