  We use molecular dynamics computer simulations to investigate the
coupling/decoupling between translational and rotational dynamics in a
glass-forming liquid of dumbbells. This is done via a careful analysis of the
$\alpha$-relaxation time $\tau_{q^{*}}^{\rm C}$ of the incoherent
center-of-mass density correlator at the structure factor peak, the
$\alpha$-relaxation time $\tau_{2}$ of the reorientational correlator, and the
translational ($D_{t}$) and rotational ($D_{r}$) diffusion constants. We find
that the coupling between the relaxation times $\tau_{q^{*}}^{\rm C}$ and
$\tau_{2}$ increases with decreasing temperature $T$, whereas the coupling
decreases between the diffusivities $D_{t}$ and $D_{r}$. In addition, the
$T$-dependence of $D_{t}$ decouples from that of $1/\tau_{2}$, which is
consistent with previous experiments and has been interpreted as a signature of
the "translation-rotation decoupling." We trace back these apparently
contradicting observations to the dynamical heterogeneities in the system. We
show that the decreasing coupling in the diffusivities $D_{t}$ and $D_{r}$ is
only apparent due to the inadequacy of the concept of the rotational diffusion
constant for describing the reorientational dynamics in the supercooled state.
We also argue that the coupling between $\tau_{q^{*}}^{\rm C}$ and $\tau_{2}$
and the decoupling between $D_{t}$ and $1/\tau_{2}$, both of which strengthen
upon cooling, can be consistently understood in terms of the growing dynamic
length scale.
