  Nonlinear generalizations of teleparallel gravity entail the modification of
a Lagrangian that is pseudoinvariant under local Lorentz transformations of the
tetrad field. This procedure consequently leads to the loss of the local
pseudoinvariance and the appearance of additional degrees of freedom (d.o.f.).
The constraint structure of f(T) gravity suggests the existence of one extra
d.o.f. when compared with GR, which should describe some aspect of the
orientation of the tetrad. The purpose of this article is to better understand
the nature of this extra d.o.f. by means of a toy model that mimics essential
features of f(T) gravity. We find that the nonlinear modification of a
Lagrangian L possessing a local rotational pseudoinvariance produces two types
of solutions. In one case the original gauge-invariant variables -- the
analogue of the metric in teleparallelism -- evolve like when governed by the
(nondeformed) Lagrangian L; these solutions are characterized by a (selectable)
constant value of its Lagrangian, which is the manifestation of the extra
d.o.f. In the other case, the solutions do contain new dynamics for the
original gauge-invariant variables, but the extra d.o.f. does not materialize
because the Lagrangian remains invariant on-shell. Coming back to f(T) gravity,
the first case includes solutions where the torsion scalar T is a constant, to
be chosen at the initial conditions (extra d.o.f.), and no new dynamics for the
metric is expected. The latter case covers those solutions displaying a genuine
modified gravity; T is not a constant, but it is (on-shell) invariant under
Lorentz transformations depending only on time. Both kinds of f(T) solutions
are exemplified in a flat FLRW universe. Finally, we present a toy model for a
higher-order Lagrangian with rotational invariance [analogous to f(R) gravity]
and derive its constraint structure and number of d.o.f.
