  The comprehension of the intricate structure associated to the local
symmetries encoded in the tetrad field, as well as its physical meaning, is
perhaps the most important unsolved problem within $f(T)$ gravity. This is
inextricably connected to the number, nature and potential impact that the
additional degree/s of freedom might have within these --and other closely
related--models of gravity in which the local Lorentz invariance is broken at
some level. Here we review and further explain some recent results which make
use of the more placid scenery provided by 2D-torsional models of gravity,
where the local symmetries adapted to a given geometry can be fully
characterized.
