Degrees of Freedom in modified Teleparallel Gravity

I discuss the issue of degrees of freedom in modified teleparallel gravity. These theories do have an extra structure on top of the usual (pseudo)Riemannian manifold, that of a flat parallel transport. This structure is absolutely abstract and unpredictable (pure gauge) in GR-equivalent models, however it becomes physical upon modifications. The problem is that, in the most popular models, this local symmetry is broken but not stably so, hence the infamous strong coupling issues. The Hamiltonian analyses become complicated and with contradictory results. A funny point is that what we see in available linear perturbation treatments of $f(T)$ gravity is much closer to the analysis with less dynamical degrees of freedom which has got a well-known mistake in it, while the more accurate work predicts much more of dynamics than what has ever been seen up to now. I discuss possible reasons behind this puzzle, and also argue in favour of studying the most general New GR models which are commonly ignored due to suspicion of ghosts.