  Axion-like particles (ALPs) are very light, neutral, spin zero bosons
predicted by many theories which try to complete the standard model of
elementary particles. ALPs interact primarily with two photons and can generate
photon-ALP oscillations in the presence of an external magnetic field. They are
attracting increasing interest since photon-ALP oscillations produce deep
consequences in very-high-energy (VHE) astrophysics. Two hints for the
existence of an ALP have recently been proposed. In this paper, we study
another effect of the photon-ALP interaction: the change of the polarization
state of photons. In particular, we study the propagation of the photon-ALP
beam starting where photons are produced - we consider photons generated in a
galaxy cluster or in the jet of a blazar - crossing several magnetized media
(blazar jet, host galaxy, galaxy cluster, extragalactic space, Milky Way) up to
their arrival at the Earth. In the presence of photon-ALP interaction, we
analyze the final photon survival probability $P_{\gamma \to \gamma}$ and the
corresponding photon degree of linear polarization $\Pi_L$ for energies in the
range $(1-10^{15}) \, \rm eV$. We observe that photons, which are expected as
unpolarized in the absence of ALPs, are made partially polarized by photon-ALP
interaction. Our findings can be tested by observatories like IXPE (already
operative), and by the proposed missions eXTP, XL-Calibur, NGXP and XPP in the
X-ray band and by COSI (approved to launch), e-ASTROGAM and AMEGO in the
high-energy range. We also discover a peculiar feature in the VHE band, where
photons at energies above $ \sim (1-10) \, \rm TeV$ are fully polarized because
of photon-ALP interaction. A possible detection of this feature would represent
a proof for the existence of an ALP, but, unfortunately, current technologies
do not allow yet to detect photon polarization up to so high energies.
