  The stellar surface density within the inner 1 kpc ($\Sigma_{1}$) has become
a popular tool for understanding the growth of galaxies and its connection with
the quenching of star formation. The emerging picture suggests that building a
central dense core is a necessary condition for quenching. However, it is not
clear whether changes in $\Sigma_{1}$ trace changes in stellar kinematics and
the growth of dispersion-dominated bulges. In this paper, we combine imaging
from the Sloan Digital Sky Survey with stellar kinematics from the Sydney-AAO
Multi-object Integral-field unit (SAMI) and Mapping Nearby Galaxies at Apache
Point Observatory (MaNGA) surveys to quantify the correlation between
$\Sigma_{1}$ and the proxy for stellar spin parameter within one effective
radius ($\lambda_{re}$) for 1599 nearby galaxies. We show that, on the
star-forming main sequence and at fixed stellar mass, changes in $\Sigma_{1}$
are mirrored by changes in $\lambda_{re}$. While forming stars, main sequence
galaxies remain rotationally-dominated systems, with their $\Sigma_{1}$
increasing but their stellar spin staying either constant or slightly
increasing. The picture changes below the main sequence, where $\Sigma_{1}$ and
$\lambda_{re}$ are no longer correlated. Passive systems show a narrower range
of $\Sigma_{1}$, but a wider range of $\lambda_{re}$ compared to star-forming
galaxies. Our results indicate that, from a structural point of view, passive
galaxies are a more heterogeneous population than star-forming systems, and may
have followed a variety of evolutionary paths. This also suggests that, if
dispersion-dominated bulges still grow significantly at $z\sim$0, this
generally takes place during, or after, the quenching phase.
