We propose a novel prior for variational 3D reconstruction that favors symmetric solutions when dealing with noisy or incomplete data. We detect symmetries from incomplete data while explicitly handling unexplored areas to allow for plausible scene completions. The set of detected symmetries is then enforced on their respective support domain within a variational reconstruction framework. This formulation also handles multiple symmetries sharing the same support. The proposed approach is able to denoise and complete surface geometry and even hallucinate large scene parts. We demonstrate in several experiments the benefit of harnessing symmetries when regularizing a surface.