While the phenomenon of birefringence is well-established in the case of visible radiation and is exploited in many fields (e.g., through the use of the polarizing optical microscope), the analogous phenomenon for X-rays has been a virtually neglected topic. Here, we demonstrate the scope and potential for exploiting X-ray birefringence to determine the orientational properties of specific types of bonds in solids. Specifically, orientational characteristics of C-Br bonds in the bromocyclohexane/thiourea inclusion compound are elucidated from X-ray birefringence measurements at energies close to the bromine K-edge, revealing inter alia the changes in the orientational distribution of the C-Br bonds associated with a low-temperature order-disorder phase transition. From fitting a theoretical model to the experimental data, reliable quantitative information on the orientational properties of the C-Br bonds is determined. The experimental strategy reported here represents the basis of a new approach for gaining insights into the orientational properties of molecules in anisotropic materials.