We consider the task of estimating the 3D orientation of an object of known category given an image of the object and a bounding box around it. Recently, CNN-based regression and classification methods have shown significant performance improvements for this task. This paper proposes a new CNN-based approach to monocular orientation estimation that advances the state of the art in four different directions. First, we take into account the Riemannian structure of the orientation space when designing regression losses and nonlinear activation functions. Second, we propose a mixed Riemannian regression and classification framework that better handles the challenging case of nearly symmetric objects. Third, we propose a data augmentation strategy that is specifically designed to capture changes in 3D orientation. Fourth, our approach leads to state-of-the-art results on the PASCAL3D+ dataset.