  We continue the exploration of the consistency of a modified-gravity theory
that generalizes General Relativity by including a dynamical torsion in
addition to the dynamical metric. The six-parameter theory we consider was
found to be consistent around arbitrary torsionless Einstein backgrounds, in
spite of its containing a (notoriously delicate) massive spin-2 excitation. At
zero bare cosmological constant, this theory was found to admit a
self-accelerating solution whose exponential expansion is sustained by a
non-zero torsion background. The scalar-type perturbations of the latter
torsionfull self-accelerating solution were recently studied and were found to
preserve the number of propagating scalar degrees of freedom, but to exhibit,
for some values of the torsion background some exponential instabilities (of a
rather mild type). Here, we study the tensor-type and vector-type perturbations
of the torsionfull self-accelerating solution, and of its deformation by a
non-zero bare cosmological constant. We find strong, "gradient" instabilities
in the vector sector. No tuning of the parameters of the theory can kill these
instabilities without creating instabilities in the other sectors. Further work
is needed to see whether generic torsionfull backgrounds are prone to
containing gradient instabilities, or if the instabilities we found are mainly
due to the (generalized) self-accelerating nature of the special de Sitter
backgrounds we considered.
