Half-jokingly one could say that what's conserved under the transformations of Lorentz group of the SME is the behaviour of the Standard Model under changes to

Half-jokingly one could say that what's conserved under the transformations of Lorentz group of the SME is the behaviour of the Standard Model under changes to the physics of gravity. A bit less jokingly, that corresponds fairly well to conserving the microscopic description of the centre of mass-energy-momentum rather than just the centre of mass-energy-momentum (at t=0 for some total energy) as in phys.se answer in the GP comment.