CT and MRI are two of the most informative modalities in spinal diagnostics and treatment planning. CT is useful when analysing bony structures, while MRI gives information about the soft tissue. Thus, fusing the information of both modalities can be very beneficial. Registration is the first step for this fusion. While the soft tissues around the vertebra are deformable, each vertebral body is constrained to move rigidly. We propose a weakly-supervised deep learning framework that preserves the rigidity and the volume of each vertebra while maximizing the accuracy of the registration. To achieve this goal, we introduce anatomy-aware losses for training the network. We specifically design these losses to depend only on the CT label maps since automatic vertebra segmentation in CT gives more accurate results contrary to MRI. We evaluate our method on an in-house dataset of 167 patients. Our results show that adding the anatomy-aware losses increases the plausibility of the inferred transformation while keeping the accuracy untouched.