  We consider the two most studied proposals of relativity theories with
observer-independent scales of both velocity and length/mass: the one discussed
by Amelino-Camelia as illustrative example for the original proposal
(gr-qc/0012051) of theories with two relativistic invariants, and an
alternative more recently proposed by Magueijo and Smolin (hep-th/0112090). We
show that these two relativistic theories are much more closely connected than
it would appear on the basis of a naive analysis of their original
formulations. In particular, in spite of adopting a rather different formal
description of the deformed boost generators, they end up assigning the same
dependence of momentum on rapidity, which can be described as the core feature
of these relativistic theories. We show that this observation can be used to
clarify the concepts of particle mass, particle velocity, and
energy-momentum-conservation rules in these theories with two relativistic
invariants.
