  We show possible transverse-momentum-dependent parton distribution functions
(TMDs) for spin-1 hadrons including twist-3 and 4 functions in addition to the
leading twist-2 ones by investigating all the possible decomposition of a quark
correlation function in the Lorentz-invariant way. The Hermiticity and parity
invariance are imposed in the decomposition; however, the time-reversal
invariance is not used due to an active role of gauge links in the TMDs.
Therefore, there exist time-reversal-odd functions in addition to the
time-reversal even ones in the TMDs. We list all the functions up to twist-4
level because there were missing terms associated with the lightcone vector $n$
in previous works on the twist-2 part and there was no correlation-function
study in the twist-3 and 4 parts for spin-1 hadrons. We show that 40 TMDs exist
in the tensor-polarized spin-1 hadron in twists 2, 3, and 4. Some expressions
of twist-2 structure functions are modified from previous derivations due to
the new terms with $n$, and we find 30 new structure functions in twists 3 and
4 in this work. Since time-reversal-odd terms of the collinear correlation
function should vanish after integrals over the partonic transverse momentum,
we obtain new sum rules for the time-reversal-odd structure functions, $ \int
d^2 k_T h_{LT}= \int d^2 k_T h_{LL} = \int d^2 k_T h_{3LL} =0$. In addition, we
indicate that new transverse-momentum-dependent fragmentation functions exist
in tensor-polarized spin-1 hadrons. The TMDs are rare observables to find
explicit color degrees of freedom in terms of color flow, which cannot be
usually measured because the color is confined in hadrons.
