  If dark energy (DE) couples to neutrinos, then there may be apparent
violations of Lorentz/CPT invariance in neutrino oscillations. The DE-induced
Lorentz/CPT violation takes a specific form that introduces neutrino
oscillations that are energy independent, differ for particles and
antiparticles, and can lead to novel effects for neutrinos propagating through
matter. We show that ultra-high-energy neutrinos may provide one avenue to seek
this type of Lorentz/CPT violation in \nu_\mu-\nu_\tau oscillations, improving
the current sensitivity to such effects by seven orders of magnitude.
Lorentz/CPT violation in electron-neutrino oscillations may be probed with the
zenith-angle dependence for high-energy atmospheric neutrinos. The ``smoking
gun,'' for DE-neutrino coupling would, however, be a dependence of neutrino
oscillations on the direction of the neutrino momentum relative to our peculiar
velocity with respect to the CMB rest frame. While the amplitude of this
directional dependence is expected to be small, it may nevertheless be worth
seeking in current data and may be a target for future neutrino experiments.
