  Clean two-dimensional electron systems in GaAs/AlGaAs heterostructures
exhibit anisotropic collective phases, the quantum Hall nematics, at high
Landau level occupancy and low temperatures. An as yet unknown native
symmetry-breaking potential consistently orients these phases relative to the
crystalline axes of the host material. Here we report an extensive set of
measurements examining the role of the structural symmetries of the
heterostructure in determining the orientation of the nematics. In single
quantum well samples we find that neither the local symmetry of the confinement
potential nor the distance between the electron system and the sample surface
dictates the orientation of the nematic. In remarkable contrast, for
two-dimensional electrons confined at a single heterointerface between GaAs and
AlGaAs, the nematic orientation depends on the depth of the two-dimensional
electron system beneath the sample surface.
