We investigate electron transport through an antidot embedded in a narrow strip of a two-dimensional topological insulator. We focus on the most generic and experimentally relevant case with broken axial spin symmetry. Spin-nonconservation allows additional scattering processes, which change the transport properties profoundly. We start from an analytical model for noninteracting transport, which we also compare with a numerical tight-binding simulation. We then extend this model by including Coulomb repulsion on the antidot, and we study the transport in the Coulomb-blockade limit. We investigate sequential tunneling and cotunneling regimes, and we find that the current-voltage characteristic allows a spectroscopic measurement of the edge-state spin textures.