A Proposed Möbius Accelerator

Any existing circular accelerator can be converted inexpensively for ``M\"obius operation'' by introducing one ``twist'' element that interchanges horizontal and vertical betatron oscillations on each particle passage. Two, not one, traversals of the ring are required to return to a corresponding state. The ring exhibits properties different from and, in important ways, superior to the original. Beam brightness can be increased while preserving large amplitude stability, and the (necessarily round) beams are robust against beam-beam interaction in colliding beam operation.