  Missions like NASA's Imaging X-ray Polarimetry Explorer (IXPE) are poised to
provide an unprecedented view of the Universe in polarized X-rays. Polarization
probes physical anisotropies, a fact exploited by particle physicists to look
for the anisotropic $a\boldsymbol{E}\cdot\boldsymbol{B}$ operator in the
axion-like-particle (ALP) Lagrangian. Such studies have typically focused on
polarization in the radio and microwaves, through local or cosmic birefringence
effects. To such polarization studies we add X-rays emanating from magnetars --
a class of neutron stars with near-critical strength magnetic fields -- that
are important targets for IXPE. ALPs produced in the neutron star core convert
to X-rays in the magnetosphere; such X-rays are polarized along the direction
parallel to the dipolar magnetic field at the point of conversion. We develop
the full theoretical formalism for ALP-induced polarization in the presence of
dipolar magnetic fields. For uncorrelated photon and ALP production mechanisms,
we completely disentangle the ALP contributions to the Stokes parameters in
terms of the ALP intensity, the ALP-to-photon conversion probability, and the
ALP-induced birefringence. In the proper limit, our results demonstrate that
the inclusion of ALPs suppresses the observed degree of circular polarization
compared to its pure astrophysical value. Our results can also be used to
impose limits on ALP couplings with IXPE polarization data from magnetars 4U
0142+61 and 1RXS J170849.0-400910, the subject of upcoming work.
