  The interplay between quantum Hall ordering and spontaneously broken
"internal" symmetries in two-dimensional electron systems with spin or
pseudospin degrees of freedom gives rise to a variety of interesting phenomena,
including novel phases, phase transitions, and topological excitations. Here we
develop a theory of broken-symmetry quantum Hall states, applicable to a class
of multi-valley systems, where the symmetry at issue is a point group element
that combines a spatial rotation with a permutation of valley indices. The
anisotropy of the dispersion relation, generally present in such systems,
favors states where all electrons reside in one of the valleys. In a clean
system, the valley "pseudo-spin" ordering, or spatial nematic ordering, occurs
via a finite temperature transition. In weakly disordered systems, domains of
pseudo-spin polarization are formed, which prevents macroscopic valley and
nematic ordering; however, the resulting state still asymptotically exhibits
the QHE. We discuss the transport properties in the ordered and disordered
regimes, and the relation of our results to recent experiments in AlAs.
