  In the context of modified teleparallel gravity, we study the generation of
primordial density fluctuations in a general scalar-torsion theory whose
Lagrangian density is an arbitrary function $f(T,\phi)$ of the torsion scalar
$T$ and a scalar field $\phi$, plus the kinetic term of this latter. It is well
known that generic modifications of teleparallel gravity are not invariant
under six-parameter local Lorentz transformations. In order to restore the
local Lorentz symmetry, we have incorporated six additional degrees of freedom
in the form of Goldstone modes of the symmetry breaking through a Lorentz
rotation of the tetrad field. After integrating out all the auxiliary modes, we
obtain a second order action for the scalar and tensor propagating modes and
their power spectrum generated during inflation. It is found that an explicit
mass term emerges in the second order action for curvature perturbation,
describing the imprints of local Lorentz violation at first-order of slow-roll.
We show that only inflationary models with nonminimal coupling functions
$f(T,\phi)$ which are non-linear in $T$, including the case of $f(T)$ gravity
with minimally coupled scalar field, can generate primordial fluctuations. For
a concrete model of inflation, we study the power-law potential by using the
latest Planck data.
