Following the first successful high-resolution study of the "perpendicular" band of dimethylsulfoxyde at 324 cm(-1), associated with the ν23 bending vibration, we discovered a sequence of fourfold degenerate clusters of rotational levels at high angular momenta J>40. This unusual system of localized states corresponds to classical rotations about a pair of "tilted" axes, which become stationary at high J, after the principal axis of inertia A loses stability and bifurcates for J≈27.