  Recent work has shown that dispersion relations with Planck scale Lorentz
violation can produce observable effects at energies many orders of magnitude
below the Planck energy M. This opens a window on physics that may reveal
quantum gravity phenomena. It has already constrained the possibility of Planck
scale Lorentz violation, which is suggested by some approaches to quantum
gravity. In this work we carry out a systematic analysis of reaction
thresholds, allowing unequal deformation parameters for different particle
dispersion relations. The thresholds are found to have some unusual properties
compared with standard ones, such as asymmetric momenta for pair creation and
upper thresholds. The results are used together with high energy observational
data to determine combined constraints. We focus on the case of photons and
electrons, using vacuum Cerenkov, photon decay, and photon annihilation
processes to determine order unity constraints on the parameters controlling
O(E/M) Lorentz violation. Interesting constraints for protons (with photons or
pions) are obtained even at O((E/M)^2), using the absence of vacuum Cerenkov
and the observed GZK cutoff for ultra high energy cosmic rays. A strong
Cerenkov limit using atmospheric PeV neutrinos is possible for O(E/M)
deformations provided the rate is high enough. If detected, ultra high energy
cosmological neutrinos might yield limits at or even beyond O((E/M)^2).
