Given a twist map on the torus, we prove that the non-existence of a rotational invariant circle implies that the shear-rotation interval is non-trivial, i.e. the existence of orbits with non-zero mean acceleration. Under the same assumption, we prove the existence of infinitely many periodic orbits and positive entropy invariant measures with non-zero shear-rotation number. The method is variational, but relies on topological understanding of the shear-rotation interval of torus homeomorphisms homotopic to a Dehn twist.