  We present a covariant formulation and model calculations of transverse
momentum-dependent quark distribution functions (TMDs) for spin-one hadrons.
Emphasis is placed on a description of these 3-dimensional distribution
functions which is independent of any constraints on the spin quantization
axis. We apply our covariant spin description to all nine leading-twist
time-reversal even $\rho$ meson TMDs in the framework provided by the
Nambu--Jona-Lasinio model, incorporating important aspects of quark confinement
via the infrared cut-off in the proper-time regularization scheme. In
particular, the behavior of the 3-dimensional TMDs in a tensor polarized
spin-one hadron are illustrated. Sum rules and positivity constraints are
discussed in detail. Results of particular interest include the finding that
the tensor polarized TMDs -- associated with spin-one hadrons -- are very
sensitive to quark orbital angular momentum, and that the TMDs associated with
the quark operator $\gamma^+\boldsymbol{\gamma}_T\gamma_5$ would vanish were it
not for dynamical chiral symmetry breaking. In addition, we find that 44% of
the $\rho$ meson's spin is carried by the orbital angular momentum of the
quarks, and that the magnitude of the tensor polarized quark distribution
function is about 30% of the unpolarized quark distribution. A qualitative
comparison between our results for the tensor structure of a quark-antiquark
bound state is made to previous experimental and theoretical results for the
two-nucleon (deuteron) bound state.
