A reanalysis of collinear factorization for inclusive Deep Inelastic Scattering shows that a novel, non perturbative spin-flip term associated with the invariant mass of the produced hadrons couples, at large enough Bjorken $x_B$, to the target's transversity distribution function. The resulting new contribution to the $g_2$ structure function can potentially explain the discrepancy between recent calculations and fits of this quantity. The new term also breaks the Burkhardt-Cottingham sum rule, now featuring an interplay between the $g_2$ and $h_1$ functions that calls for a re-examination of their small-$x$ behavior. As part of the calculation leading to these results, a new set of TMD sum rules is derived by relating the single-hadron quark fragmentation correlator to the fully dressed quark propagator by means of integration over the hadronic momenta and spins. A complete set of momentum sum rules is obtained for transverse-momentum-dependent quark fragmentation functions up to next-to-leading twist.