Recently, two fundamental topological properties of a magnetic vortex at the interface of a superconductor (SC) and a strong topological insulator (TI) have been established: The vortex carries both a Majorana zero mode relevant for topological quantum computation and, for a time-reversal invariant TI, a charge of e/4. This fractional charge is caused by the axion term in the electromagnetic Lagrangian of the TI. Here we determine the angular momentum J of the vortices, which in turn determines their mutual statistics. Solving the axion-London electrodynamic equations including screening in both a SC and a TI, we find that the elementary quantum of angular momentum of the vortex is -n2/8, where n is the flux quantum of the vortex line. Exchanging two elementary fluxes thus changes the phase of the wave function by -π/4.