After a decade of near stability at P=0.146 d, the photometric "superhump" periodicity of the old nova V603 Aquilae experienced a remarkable change between 1991 and 1992. Observation in 1992--1994 indicates that the dominant signal was then at a period in the range 0.1338-0.1345 d, 3% shorter than the orbital period. Like its predecessor, the new signal also wanders in period on a timescale of a few months. The full amplitude in 1994 was 0.20 mag, more than twice as great as the superhump displayed during 1980--1991. An intensive observing campaign in 1994 revealed that the old superhump at 0.146 d was still present with approximately undiminished amplitude (averaging 0.07 mag). In a precession model, the simultaneous presence of superhumps above and below the orbital period strongly suggests identification with two independent types of precessional sideband. The observed periods and period changes are consistent with a simple hypothesis: that the longer period ("positive superhump") arises from the prograde motion of the line of apsides, and the shorter period ("negative superhump") arises from the retrograde motion of the line of nodes. A detailed account of how a fluid disk manages to maintain such well-organized motions is sorely needed.