Ever since the work of von Ignatowsky circa 1910 it has been known (if not always widely appreciated) that the relativity principle, combined with the basic and fundamental physical assumptions of locality, linearity, and isotropy, leads almost uniquely to either the Lorentz transformations of special relativity or to Galileo's transformations of classical Newtonian mechanics. Consequently, if one wishes (for whatever reason) to entertain the possibility of Lorentz symmetry breaking within the context of the class of local physical theories, then it seems likely that one will have to abandon (or at the very least grossly modify) the relativity principle. Working within the framework of local physics, we reassess the notion of spacetime transformations between inertial frames in the absence of the relativity principle, arguing that significant and nontrivial physics can still be extracted as long as the transformations are at least linear. An interesting technical aspect of the analysis is that the transformations now form a groupoid/pseudo-group - it is this technical point that permits one to evade the von Ignatowsky argument. Even in the absence of a relativity principle we can (assuming locality and linearity) nevertheless deduce clear and compelling rules for the transformation of space and time, rules for the composition of 3-velocities, and rules for the transformation of energy and momentum. Within this framework, the energy-momentum transformations are in general affine, but may be chosen to be linear, with the 4-component vector P = (E,-p T ) transforming as a row vector, while the 4-component vector of space-time position X = (t, x T ) T transforms as a column vector. As part of the analysis we identify two particularly elegant and physically compelling models implementing "minimalist" violations of Lorentz invariance - in the first of these minimalist models all Lorentz violations are confined to carefully delineated particle physics sub-sectors, while the second minimalist Lorentz-violating model depends on one free function of absolute velocity, but otherwise preserves as much as possible of standard Lorentz invariant physics.