We present a fully relativistic calculation of the bremsstrahlung emitted by twisted electrons propagating in the field of bare heavy nuclei. The electron-nucleus interaction is accounted for to all orders in the nuclear binding strength parameter
αZ, thus allowing us to investigate the bremsstrahlung in a strong field, where the effects of the “twistedness” are expected to be most pronounced. To explore these effects, we study the angular and polarization properties of the photons emitted in course of the inelastic twisted electrons scattering by the gold target. The influence of the kinematic parameters of the incident electrons on the double-differential cross section and the degree of the linear polarization is also discussed.