We study, within an effective field theory framework, $O(E^{2}/\Mpl^{2})$
Planck-scale suppressed Lorentz invariance violation (LV) effects in the
neutrino sector, whose size we parameterize by a dimensionless parameter
$\eta_{\nu}$. We find deviations from predictions of Lorentz invariant physics
in the cosmogenic neutrino spectrum. For positive O(1) coefficients no neutrino
will survive above $10^{19} \eV$. The existence of this cutoff generates a bump
in the neutrino spectrum at energies of $10^{17} \eV$. Although at present no
constraint can be cast, as current experiments do not have enough sensitivity
to detect ultra-high-energy neutrinos, we show that experiments in construction
or being planned have the potential to cast limits as strong as $\eta_{\nu}
\lesssim 10^{-4}$ on the neutrino LV parameter, depending on how LV is
distributed among neutrino mass states. Constraints on $\eta_{\nu} < 0$ can in
principle be obtained with this strategy, but they require a more detailed
modeling of how LV affects the neutrino sector.
