  Recently the study of threshold kinematic requirements for
particle-production processes has played a very significant role in the
phenomenology of theories with departures from Poincare' symmetry. We here
specialize these threshold studies to the case of a class of violations of
Poincare' symmetry which has been much discussed in the literature on
Horava-Lifshitz scenarios. These involve modifications of the energy-momentum
("dispersion") relation that may be different for different types of particles,
but always involve even powers of energy-momentum in the correction terms. We
establish the requirements for compatibility with the observed cosmic-ray
spectrum, which is sensitive to the photopion-production threshold. We find
that the implications for the electron-positron pair-production threshold are
rather intriguing, in light of some recent studies of TeV emissions by Blazars.
Our findings should also provide motivation for examining the fate of the law
of energy-momentum conservation in Horava-Lifshitz-type theories.
