  Relativity theory and its underlying Lorentz and CPT invariance represent key
principles of physics and therefore require continued experimental scrutiny
across the broadest possible range of energy scales and physical systems.
Possibilities for tests of these symmetries in precision $\beta$-decay
experiments with focus on KATRIN are investigated. It is found that
countershaded departures from relativity, which represent a particular
challenge to detection, are accessible in such experiments. In this context, it
is argued that KATRIN would be in an excellent position to conduct the
first-ever experimental search for countershaded Lorentz and CPT breaking in
neutrinos.
