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.
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