  High-brilliance synchrotron radiation sources have opened new avenues for
X-ray polarization analysis that go far beyond conventional polarimetry in the
optical domain. With linear X-ray polarizers in a crossed setting polarization
extinction ratios down to 10$^{-10}$ can be achieved. This renders the method
sensitive to probe tiniest optical anisotropies that would occur, for example,
in strong-field QED due to vacuum birefringence and dichroism. Here we show
that high-purity polarimetry can be employed to reveal electronic anisotropies
in condensed matter systems with utmost sensitivity and spectral resolution.
Taking CuO and La$_2$CuO$_4$ as benchmark systems, we present a full
characterization of the polarization changes across the Cu K-absorption edge
and their separation into dichroic and birefringent contributions. At
diffraction-limited synchrotron radiation sources and X-ray lasers, where
polarization extinction ratios of 10$^{-12}$ can be achieved, our method has
the potential to assess birefringence and dichroism of the quantum vacuum in
extreme electromagnetic fields.
