  It has recently been shown that $f(T)$ gravity has $\frac{n(n-3)}{2}+1$
physical degrees of freedom (d.o.f.) in $n$ dimensions, contrary to previous
claims. The simplest physical interpretation of this fact is that the theory
possesses a scalar d.o.f. This is the case of $f(R)$ gravity, a theory that can
be understood in the Einstein frame as general relativity plus a scalaron. The
scalar field that represents the extra d.o.f. in $f(T)$ gravity encodes
information about the parallelization of the spacetime, which is detected
through a reinterpretation of the equations of motion in both the teleparallel
Jordan and Einstein frames. The trace of the equations of motion in $f(T)$
gravity shows the propagation of the scalar d.o.f., giving an accurate proof of
its existence. We also provide a simple toy model of a physical system with
rotational pseudoinvariance, like $f(T)$ gravity, which gives insights into the
physical interpretation of the extra d.o.f. We discuss some implications and
unusual features of the previously worked out Hamiltonian formalism for $f(T)$
gravity. Finally we show some mathematical tools to implement the Hamiltonian
formulation in the Einstein frame of $f(T)$ gravity, which exhibits some
problems that should be addressed in future works.
