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X-ray dichroism is observed in crystalline vanadyl bisacetylacetonate, VO2+.2(C5H7O2)-, near the vanadium K edge in absorption experiments with linearly polarized synchrotron radiation. The Kossel structure is polarized as a result of the symmetries of molecular orbitals in this strongly anisotropic molecule. The extended X-ray absorption fine structure is polarized because of the difference in distance to the axial and equatorial oxygen neighbors. In this wavelength region, the anomalous-scattering terms and the index of refraction vary with the direction of polarization. Some of the consequences and possible applications of these properties are discussed.