  In the presence of spontaneous symmetry breaking, the alignment of the vacuum
with respect to the gauge group is often controlled by quadratically divergent
operators in the low energy non-linear sigma model. In principle the magnitudes
and signs of these operators can be changed by making different assumptions
about the ultraviolet physics, but in practice all known ways of regulating
these theories preserve the naive vacuum alignment. We show that by
"integrating in" different sets of heavy spin-one fields, it is possible to UV
extend certain non-linear sigma models into two distinct UV insensitive
theories. These UV extensions have identical low energy degrees of freedom but
different radiative potentials, making it possible to engineer two different
vacuum alignments for the original non-linear sigma model. Our construction
employs "non-square" theory spaces which generically violate the common lore
that the preferred vacuum alignment preserves the maximal gauge symmetry. By UV
extending the SO(9)/(SO(4) X SO(5)) little Higgs model, we find a radiative
potential that deviates from the naive expectation but does not stabilize the
correct vacuum for proper electroweak symmetry breaking.
