  Boosted and rotated tetrad backgrounds for the Minkowski space are studied in
$f(T)$ gravity. We perform Lorentzian perturbations at first order around
non-trivial backgrounds and show that some Lorentz modes can exhibit
non-trivial dynamics and can propagate in time. This remarkable feature gives
evidence of additional mode(s) in the Lorentzian sector which have no precedent
in Lorentz-violating modified gravities, however they can cast doubts onto even
theoretical viability of these models.
