Acquiring 3D geometry of an object is a tedious and time-consuming task, typically requiring scanning the surface from multiple viewpoints. In this work we focus on reconstructing complete geometry from a single scan acquired with a low-quality consumer-level scanning device. Our method uses a collection of example 3D shapes to build structural part-based priors that are necessary to complete the shape. In our representation, we associate a local coordinate system to each part and learn the distribution of positions and orientations of all the other parts from the database, which implicitly also defines positions of symmetry planes and symmetry axes. At the inference stage, this knowledge enables us to analyze incomplete point clouds with substantial occlusions, because observing only a few regions is still sufficient to infer the global structure. Once the parts and the symmetries are estimated, both data sources, symmetry and database, are fused to complete the point cloud. We evaluate our technique on a synthetic dataset containing 481 shapes, and on real scans acquired with a Kinect scanner. Our method demonstrates high accuracy for the estimated part structure and detected symmetries, enabling higher quality shape completions in comparison to alternative techniques.