  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 K$\alpha_1$ ($1s 2p_{3/2} ^{1,3}P_{1,2} \to 1s^2 ^1S_0$) 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 K$\alpha_1$
line can be significantly affected by the mutipole mixing between the leading
$M2$ and hyperfine-induced $E1$ components of $1s2p ^3P_2, F_i \to 1s^2 ^1S_0,
F_f$ transitions. This $E1$-$M2$ mixing strongly depends on the nuclear
properties of the considered isotopes and can be addressed experimentally at
existing heavy-ion storage rings.
