  In the literature on $f(T)$ gravity, the status of local Lorentz invariance
and the number of physical degrees of freedom have been controversial issues.
Relying on a detailed Hamiltonian analysis, we show that there are several
scenarios describing how local Lorentz invariance can be broken, but in the
generic case, the number of physical degrees of freedom is found to be $N^*=5$;
in $D$ dimensions, this number is $N^*=D(D-3)/2+(D-1)$. As expected, the theory
is vulnerable to having problematical propagating modes. We compare our results
with those existing in the literature. As a byproduct of our analysis, the
diffeomorphysm invariance is explicitly confirmed.
