  Polarization measurements of thermal radiation from magnetic white dwarf
(MWD) stars have been proposed as a probe of axion-photon mixing. The radiation
leaving the surface of the MWD is unpolarized, but if low-mass axions exist
then photons polarized parallel to the direction of the MWD's magnetic field
may convert into axions, which induces a linear polarization dependent on the
strength of the axion-photon coupling $g_{a\gamma\gamma}$. We model this
process by using the formalism of axion-photon mixing in the presence of
strong-field vacuum birefringence to show that of all stellar types MWDs are
the most promising targets for axion-induced polarization searches. We then
consider linear polarization data from multiple MWDs, including SDSS J135141
and Grw+70$^\circ$8247, to show that after rigorously accounting for
astrophysical uncertainties the axion-photon coupling is constrained to
$|g_{a\gamma\gamma}| \lesssim 5.4 \times 10^{-12}$ GeV$^{-1}$ at 95% confidence
for axion masses $m_a \lesssim 3 \times 10^{-7}$ eV. This upper limit puts in
tension the previously-suggested explanation of the anomalous transparency of
the Universe to TeV gamma-rays in terms of axions. We identify MWD targets for
which future data and modeling efforts could further improve the sensitivity to
axions.
