[math-fun] flat corners

Here's a puzzle for which I found a nice but degenerate solution: Find a toroidal polyhedron of genus 1 whose corners are all "flat" in the sense that the angles of the faces meeting at each corner add up to 360 degrees. (I agree that there a number of inequivalent ways to interpret the problem, but I don't know how to clarify the statement of the problem without giving too big a hint!) I'm curious whether equally nice but less degenerate examples are known. Jim