  We show transverse-momentum-dependent parton distribution functions (TMDs)
for spin-1 hadrons including twist-3 and 4 functions by taking the
decomposition of a quark correlation function in the Lorentz-invariant way with
the conditions of Hermiticity and parity invariance. We found 30 new TMDs in
the tensor-polarized spin-1 hadron at twists 3 and 4 in addition to 10 TMDs at
twist 2. Since time-reversal-odd terms of the collinear correlation function
should vanish after integrals over the partonic transverse momentum, we
obtained new sum rules for the time-reversal-odd structure functions, $\bm{\int
d^2 k_T g_{LT}
  = \int d^2 k_T h_{LL} = \int d^2 k_T h_{3LL}}$$\bm{=0}$, at twists 3 and 4.
We also indicated that transverse-momentum-dependent fragmentation functions
exist in tensor-polarized spin-1 hadrons. The TMDs can probe color degrees of
freedom, so that they are valuable in providing unique opportunities for
creating interdisciplinary physics fields such as gluon condensate, color
Aharonov-Bohm effect, and color entanglement. We also found three new collinear
PDFs at twists 3 and 4, and a twist-2 relation and a sum rule were derived in
analogy to the Wandzura-Wilczek relation and the Burkhardt-Cottingham sum rule
on the structure function $\bm{g_2}$.
