  A generalized idea of gauge invariance, that embodies into the Wilson lines
the spin-dependent Pauli term $\sim F^{\mu\nu}[\gamma_\mu, \gamma_\nu]$, is
applied to set up a new framework for the operator definition of
transverse-momentum-dependent parton densities (TMDs). We show that such a
treatment of gauge invariance is justified, since it does not change the
leading-twist behavior of the TMDs, albeit it contributes to their twist-three
properties, in particular, to their anomalous dimensions. We discuss other
consequences of this generalization and its possible applications to lattice
simulations of the TMDs.
