A type of two-mirror telescope is described in which light leaving the secondary mirror is allowed to reflect off the primary mirror again before coming to the final focus. It is shown that this double bounce off the primary mirror results in a telescope that may be configured to be corrected for spherical aberration, coma, and astigmatism. In fact, there are four distinct families of designs that are anastigmatic, encompassing a wide variety of optical forms. The more interesting designs are described in detail, and some performance numbers are given.