In the standard refraction routine, a Jackson cross-cylinder is reversed in front of the eye by the process of flipping it about its handle. Reverse telescopes have applications in low vision. In ophthalmoscopy, the practitioner views a patient's retina using light that traverses the eye in a direction opposite to that involved in vision. These represent examples of flipped systems that are involved in optometry. The purpose of this paper is to examine what happens to the optical character of an arbitrary optical system when it is flipped in this manner. The analysis is in terms of the ray transference of the system and, because the transference completely characterises the first-order optics of a system, is complete within the limitations of linear optics. It allows for elements in the system to be astigmatic and decentred. An expression is derived for the transference of the flipped system in terms of the transference of the system itself. Expressions are also obtained for the fundamental first-order optical properties and the dioptric power of the flipped system. Three numerical examples are given in the Appendix.