  Axion-like-particles (ALPs) produced in the core of a magnetar can convert to
photons in the magnetosphere, giving rise to novel features in the X-ray
spectrum. Since ALPs only mix with the parallel mode of the photon, the
polarization of the soft and hard X-ray spectra is predicted to have an O-mode
component, in addition to the mainly X-mode component given by most
astrophysical models. The relative strength of the O-mode component depends on
the intensity of ALPs produced in the core and the probability of conversion.
We quantify our results by considering X-ray emission produced both by
astrophysical processes and by ALP-photon conversion, in an uncorrelated
fashion, and in different relative proportions, which we parametrize by the
angle $\chi_0$. We then define a normalized astrophysics-subtracted Stokes
parameter $R$ which only acquires non-zero values in the presence of ALP-photon
conversion. We find, remarkably, that the parameter $R$ factorizes into a
product of the ALP-to-photon conversion probability and $\cos(2\chi_0)$ and
display $R$, as well as the usual Stokes parameter $Q$, as a function of the
photon energy and relative fractions of ALP and photon intensities. For
benchmark points currently allowed by the CAST experiment, the O-mode
prediction can be tested in future X-ray polarimeters and used either to
constrain ALPs or find evidence for them.
