We analyze the structure of generalized off-diagonal and transverse-momentum dependent quark-quark and gluon-gluon correlators for a spin-1/2 hadron. Using the light-front formalism, we provide a parametrization in terms of the parton generalized transverse-momentum dependent distributions that emphasizes the multipole structure of the correlator. The results for the quark-quark correlation functions are consistent with an alternative parametrization given in terms of Lorentz covariant structures. The parametrization for the gluon-gluon generalized correlator is presented for the first time and allows one to introduce new correlation functions which can be relevant for phenomenological applications.