Interaction of twisted strong laser radiation with electrons in the classical regime is considered. We investigate transfer of the angular momentum of absorbed laser photons to the emitted radiation. An interaction regime is considered where radiation reaction is negligible and the formation length of radiation is comparable to or larger than the laser wavelength. The latter condition ensures that the structure of the laser field plays a role in the electron dynamics during the formation of radiation. We distinguish the case of a single electron from that of an electron beam. For a single electron, the spin angular momentum of the driving laser photons is transferred to the radiation field, while the orbital angular momentum of the laser field is not. We conclude that in the classical regime, to imprint the angular momentum of twisted light on radiation, an electron beam is a prerequisite. In the latter case, nonlinear Thomson scattering of twisted light off an ultrarelativistic electron beam produces high-frequency radiation that is twisted, with a topological charge proportional to the harmonic order.