The linear polarization of the characteristic photon emission from few-electron ions is studied for its sensitivity with regard to the nuclear spin and magnetic moment of the ions. Special attention is paid, in particular, to the Kα1 (1s2p 3/2 1,3 P1,2 → 1s 2 1 S0) decay of selected helium-like ions following the radiative electron capture into initially hydrogen-like species. Based on the density matrix theory, a unified description is developed that includes both, the many-electron and hyperfine interactions as well as the multipole-mixing effects arising from the expansion of the radiation field. It is shown that the polarization of the Kα1 line can be significantly affected by the mutipole mixing between the leading M 2 and hyperfine-induced E1 components of 1s2p 3 P2, Fi → 1s 2 1 S0, F f transitions. This E1-M 2 mixing strongly depends on the nuclear properties of the considered isotopes and can be addressed experimentally at existing heavy-ion storage rings.