Recent advances in the production of twisted electron beams with a subnanometer spot size offer unique opportunities to explore the role of orbital angular momentum in basic atomic processes. In the present work, we address one of these processes: radiative recombination of twisted electrons with bare ions. On the basis of the density matrix formalism and the non-relativistic Schrödinger theory, analytical expressions are derived for the angular distribution and the linear polarization of photons emitted due to the capture of twisted electrons into the ground state of (hydrogen-like) ions. We show that these angular and polarization distributions are sensitive to both the transverse momentum and the topological charge of the electron beam. To observe in particular the value of this charge, we propose an experiment that makes use of the coherent superposition of two twisted beams.