We demonstrate that the two-dimensional electron system in a strong perpendicular magnetic field has stable states which break rotational but not translational symmetry. The Laughlin fluid becomes unstable to these states in quantum wells whose thickness exceeds a critical value which depends on the electron density. The order parameter at 1/3 reduced density resembles that of a nematic liquid crystal, in that a residual twofold rotation axis is present in the low-symmetry phase. At filling factors 1/5 and 1/7, there are states with fourfold and sixfold axes, as well. We discuss the experimental detection of these phases.