  It is shown that the extended teleparallel gravitational theories, known as
f(T) theories, inherit some on shell local Lorentz invariance associated with
the tetrad field defining the spacetime structure. We discuss some enlightening
examples, such as Minkowski spacetime and cosmological
(Friedmann-Robertson-Walker and Bianchi type I) manifolds. In the first case,
we show that the absence of gravity reveals itself as an incapability in the
selection of a preferred parallelization at a local level, due to the fact that
the infinitesimal local Lorentz subgroup acts as a symmetry group of the frame
characterizing Minkowski spacetime. Finite transformations are also discussed
in these examples and, contrary to the common lore on the subject, we conclude
that the set of tetrads responsible for the parallelization of these manifolds
is quite vast and that the remnant group of local Lorentz transformations
includes one and two dimensional Abelian subgroups of the Lorentz group.
