  We report measurements of the quantum Hall state energy gap at avoided
crossings between Landau levels originating from different conduction band
valleys in AlAs quantum wells. These gaps exhibit an approximately linear
dependence on magnetic field over a wide range of fields and filling factors.
More remarkably, we observe an unexpected dependence of the gap size on the
relative spin orientation of the crossing levels, with parallel-spin crossings
exhibiting larger gaps than antiparallel-spin crossings.
