Inverse Compton (IC) scattering of the anisotropic CMB fluctuations off cosmic electron plasmas generates a polarization of the associated Sunyaev-Zel'dovich (SZ) effect. The polarized SZ effect has important applications in cosmology and in astrophysics of galaxy clusters. However, this signal has been studied so far mostly in the non-relativistic regime which is valid only in the very low electron temperature limit for a thermal electron population and, as such, has limited astrophysical applications. Partial attempts to extend this calculation to the IC scattering of a thermal electron plasma in the relativistic regime have been done but these cannot be applied to a more general or mildly relativistic electron distribution. In this paper we derive a general form of the SZ effect polarization that is valid in the full relativistic approach for both thermal and non-thermal electron plasmas, as well as for a generic combination of various electron population which can be co-spatially distributed in the environments of galaxy clusters or radiogalaxy lobes. We derive the spectral shape of the Stokes parameters induced by the IC scattering of every CMB multipole for both thermal and non-thermal electron populations, focussing in particular on the CMB quadrupole and octupole that provide the largest detectable signals in cosmic structures (like galaxy clusters). We found that the CMB quadrupole induced Stoke parameter Q is always positive with a maximum amplitude at a frequency approximate to 216 GHz which increases non-linearly with increasing cluster temperature. On the contrary, the CMB octupole induced Q spectrum shows a cross-over frequency which depends on the cluster electron temperature in a linear way, while it shows a non-linear dependence on the minimum momentum p1 of a non-thermal power-law spectrum as well as a linear dependence on the power-law spectral index of the non-thermal electron population. We discuss some of the possibilities to disentangle the quadrupole-induced Q spectrum from the octupole-induced one which will allow to measure these important cosmological quantities through the SZ effect polarization at different cluster locations in the universe. We finally apply our model to the Bullet cluster and derive the visibility windows of the total, quandrupole-induced and octupole-induced Stoke parameter Q in the frequency ranges accessible to SKA, ALMA, MILLIMETRON and CORE++ experiments.
