We report the discovery in the Sloan Digital Sky Survey (SDSS) and the SDSS-III Baryon Oscillation Spectroscopic Survey of 17 broad absorption line (BAL) quasars with high-ionization troughs that include absorption redshifted relative to the quasar rest frame. The redshifted troughs extend to velocities up to v ≃ 12 000 km s-1 and the trough widths exceed 3000 km s-1 in all but one case. Approximately 1 in 1000 BAL quasars with blueshifted C IV absorption also has redshifted C IV absorption; objects with C IV absorption present only at redshifted velocities are roughly four times rarer. In more than half of our objects, redshifted absorption is seen in C II or Al III as well as C IV, making low-ionization absorption at least 10 times more common among BAL quasars with redshifted troughs than among standard BAL quasars. However, the C IV absorption equivalent widths in our objects are on average smaller than those of standard BAL quasars with low-ionization absorption. We consider several possible ways of generating redshifted absorption. The two most likely possibilities may be at work simultaneously, in the same objects or in different ones. Rotationally dominated outflows seen against a quasar's extended continuum source can produce redshifted and blueshifted absorption, but variability consistent with this scenario is seen in only one of the four objects with multiple spectra. The infall of relatively dense and low-ionization gas to radii as small as 400 Schwarzschild radii can in principle explain the observed range of trough profiles, but current models do not easily explain the origin and survival of such gas. Whatever the origin(s) of the absorbing gas in these objects, it must be located at small radii to explain its large redshifted velocities, and thus offers a novel probe of the inner regions of quasars.