Twisted photons and electrons are freely-propagating particles that carry orbital angular momentum (OAM) along their propagation direction. These twisted particles currently attract considerable interest in both fundamental and applied research and extensive studies have been performed to explore their basic properties and potential applications. In particular, much attention has been focused on how the'twistednes' of a particle may affect its interaction with light and matter. Two fundamental processes that can provide substantial insights into this (light-matter) interaction are the photoionization of atoms or ions as well as its (time-reversed) counterpart: radiative recombination. In this work, we develop a novel theoretical formalism to study (a) the ionization of hydrogen-like ions by twisted photons and (b) the radiative capture of twisted electrons by bare ions. For both processes, special attention is paid to the properties of the emitted particles —photons and electrons—as characterized by their angular distribution or polarization state. Calculations are performed based on the non-relativistic first-order perturbation theory and the density matrix approach. The obtained results show that the OAM of the incident twisted particle beam may significantly affect the properties of the outgoing particles in both, the ionization (a) and the recombination process (b). We demonstrate that such (OAM-dependent) effects can be observed most easily in the so-called nonparaxial beam regime where the magnitude of the transverse linear momentum of the incoming beam is of the same order as the longitudinal one.