  Teleparallel gravity is a formulation of general relativity that is
physically equivalent to metric gravity if the gravitational action has the
Einstein-Hilbert form and matter is minimally coupled. However, scalar fields
generally couple directly to the connection, breaking the equivalence. In
particular, this happens for the Standard Model Higgs. We show that a
teleparallel theory with a non-minimally coupled scalar field has no linear
scalar perturbations, and therefore cannot give successful inflation, unless
the non-minimal coupling functions satisfy a particular relation. If the
relation is satisfied, Higgs inflation can give an arbitrarily large
tensor-to-scalar ratio $r$. Our results also apply to $f(T)$ theories, as they
are scalar-tensor theories written in different field coordinates. We discuss
generalisation to more complicated actions.
