Radiatively decaying dark matter may be searched through investigating the photon spectrum of galaxies and galaxy clusters. We explore whether the properties of dark matter can be constrained through the study of a polarization state of emitted photons. Starting from the basic principles of quantum mechanics we show that the models of symmetric dark matter are indiscernible by the photon polarization. However, we find that the asymmetric dark matter consisted of Dirac fermions is a source of circularly polarized photons, calling for the experimental determination of the photon state. (C) 2016 The Authors. Published by Elsevier B.V.