We propose a novel neural architecture for representing 3D surfaces, which harnesses two complementary shape representations: (i) an explicit representation via an atlas (ii) an implicit-function representation. We make these two representations synergistic by introducing novel consistency losses. Our hybrid architecture outputs results which are superior to the output of the two equivalent single-representation networks, yielding smoother explicit surfaces with more accurate normals, and a more accurate implicit occupancy function. Additionally, our surface reconstruction step can directly leverage the explicit atlas-based representation. This process is computationally efficient, and can be directly used by differentiable rasterizers, enabling training our hybrid representation with image-based losses.