The deep neural networks combined with the statistical shape model have enabled efficient deformable 2D-3D registration and recovery of 3D anatomical structures from a single radiograph. However, the recovered volumetric image tends to lack the volumetric fidelity of fine-grained anatomical structures and explicit consideration of cross-dimensional semantic correspondence. In this paper, we introduce a simple but effective solution for semantically-consistent deformable 2D-3D registration and detailed volumetric image recovery by inferring a voxel-wise registration field between the cone-beam computed tomography and a single lateral cephalometric radiograph (LC). The key idea is to refine the initial statistical model-based registration field with craniofacial structural details and semantic consistency from the LC. Specifically, our framework employs a self-supervised scheme to learn a voxel-level refiner of registration fields to provide fine-grained craniofacial structural details and volumetric fidelity. We also present a weakly supervised semantic consistency measure for semantic correspondence, relieving the requirements of volumetric image collections and annotations. Experiments showcase that our method achieves deformable 2D-3D registration with performance gains over state-of-the-art registration and radiograph-based volumetric reconstruction methods. The source code is available at https://github.com/Jyk-122/SC-DREG.