  We show that particle scattering in general curved backgrounds entails {\it
six} independent, kinematical Mandelstam-like invariants, instead of the two in
flat spacetime. Spacetime isometries are shown to lead to constraints between
these parameters, so that for standard black holes like Schwarzschild,
Reissner-Nordstr\"om, or Kerr spacetimes, the number of {\it independent}
parameters may be less than six. We compute the values of these independent
parameters very close to the event horizon of the black holes. We demonstrate
the existence of kinematical domains in the parameter space of particle
trajectories for which some of the independent invariants may become unbounded
above, as the point of collision approaches the event horizon. For particle
scattering, this would imply the possibility of scattering with very large
center-of-mass energy squared and/or very large momentum-transferred squared,
making this astrophysically a laboratory for physics beyond the standard model.
