One of the primary difficulties in treating patients with genetic syndromes is diagnosing their condition. Many syndromes are associated with characteristic facial features that can be imaged and utilized by computer-assisted diagnosis systems. In this work, we develop a novel 3D facial surface modeling approach with the objective of maximizing diagnostic model interpretability within a flexible deep learning framework. Therefore, an invertible normalizing flow architecture is introduced to enable both inferential and generative tasks in a unified and efficient manner. The proposed model can be used (1) to infer syndrome diagnosis and other demographic variables given a 3D facial surface scan and (2) to explain model inferences to non-technical users via multiple interpretability mechanisms. The model was trained and evaluated on more than 4700 facial surface scans from subjects with 47 different syndromes. For the challenging task of predicting syndrome diagnosis given a new 3D facial surface scan, age, and sex of a subject, the model achieves a competitive overall top-1 accuracy of 71%, and a mean sensitivity of 43% across all syndrome classes. We believe that invertible models such as the one presented in this work can achieve competitive inferential performance while greatly increasing model interpretability in the domain of medical diagnosis.