Inspired by the seminal result that a graph and an associated rotation system uniquely determine the topology of a closed manifold, we propose a combinatorial method for reconstruction of surfaces from points. Our method constructs a spanning tree and a rotation system. Since the tree is trivially a planar graph, its rotation system determines a genus zero surface with a single face which we proceed to incrementally refine by inserting edges to split faces. In order to raise the genus, special handles are added in a later stage by inserting edges between different faces and thus merging them. We apply our method to a wide range of input point clouds in order to investigate its effectiveness, and we compare our method to several other surface reconstruction methods. It turns out that our approach has two specific benefits over these other methods. First, the output mesh preserves the most information from the input point cloud. Second, our method provides control over the topology of the reconstructed surface. Code is available on https://github.com/cuirq3/RsR.