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 similar or equal to 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 CIV absorption also has redshifted C IV absorption; objects with CIV absorption present only at redshifted velocities are roughly four times rarer. In more than half of our objects, redshifted absorption is seen in CII or Al III as well as CIV, making low-ionization absorption at least 10 times more common among BAL quasars with redshifted troughs than among standard BAL quasars. However, the CIV absorption equivalent widths in our objects are on average smaller than those of standard BAL quasars with low-ionization absorption.