We discuss the possibility of explaining the observation of ultra-high-energy cosmic rays with energy above the GZK cutoff, saving the relativity principle and the (possibly deformed) Lorentz symmetry, as proposed recently by several authors. Since it is known that the Lie group structure of the Lorentz group cannot be deformed, we study the deformations (up to isomorphisms) of the mass-shell, considered as an abstract three-dimensional homogeneous space. We find that in the massive case the mass-shell cannot be deformed and in the massless case there are deformations, but their physical interpretation is problematic. The components of the four-momentum are considered as (redundant) coordinates on the abstract mass-shell. Reinterpreting an old result, we note that if the four-momentum is conserved its components must be the usual ones, with linear Lorentz transformation properties. Even if four-momentum is not conserved at high center-of-mass energies, the linearly transforming coordinates can always be used to describe in a convenient way the kinematics of collision processes and they satisfy the GKZ cutoff. We suggest that, if one wants to save the relativity principle, one should look for new physics in the collisions between the ultra-high-energy cosmic rays and the nuclei of the high atmosphere.