We study the hyperk\"ahler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperk\"ahler Kirwan map is surjective and produce a formula that recursively calculates the Betti numbers of these spaces for all ranks. Building on a natural analogy between hyperpolygons and parabolic Higgs bundles, we identify hyperpolygon spaces with certain degenerate Hitchin systems, and use this to establish their complete integrability, for ranks up to and including 3.