Central figure-8 cross-cuts make surfaces cylindrical

We prove: If a complete connected smooth surface M in euclidean 3-space has general position, intersects some plane along a clean figure-8 (a loop with total curvature zero) and all compact intersections with planes have central symmetry, then M is a (geometric) cylinder over some central figure-8. On the way, we establish interesting facts about centrally symmetric loops in the plane; for instance, a clean loop with even rotation number 2k can never be central unless it passes through its center exactly twice and k=0.