  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.
