We study continuity conditions for the operator of harmonic reflection of functions over boundaries of simple Carathéodory domains. This operator is considered as one acting from a space of functions of Lipschitz–Hölder type defined by a general modulus of a continuity, into another space of such kind. The results obtained are based on the continuity criterion for the Poisson operator (acting in the same spaces of functions) in the domains in question. This criterion is also obtained in the paper. These results generalize and refine those obtained in the recent work by the second author and P. Paramonov (Analysis and Mathematical Physics, 2019).