  We use accurate radial velocities for 1981 member stars in 20 Galactic
globular clusters, collected within our large survey aimed at the analysis of
the Na-O anti-correlation, to study the internal kinematics of the clusters. We
performed the first systematic exploration of the possible connections between
cluster kinematics and the multiple populations phenomenon in GCs. We did not
find any significant correlation between Na abundance and either velocity
dispersion or systemic rotation. We searched for systemic rotation in the eight
clusters of our sample that lack such analysis from previous works in the
literature (NGC2808, NGC5904, NGC6171, NGC6254, NGC6397, NGC6388, NGC6441, and
NGC6838). These clusters are found to span a large range of rotational
amplitudes, from ~0.0 km/s (NGC6397) to ~13.0 km/s (NGC6441). We found a
significant correlation between the ratio of rotational velocity to central
velocity dispersion (V_{rot}/sigma_0) and the Horizontal Branch Morphology
parameter (B-R)/(B+R+V). V_{rot}/sigma_0 is found to correlate also with
metallicity, possibly hinting to a significant role of dissipation in the
process of formation of globular clusters. V_{rot} is found to correlate well
with (B-R)/(B+R+V), M_V, sigma_0 and [Fe/H]. All these correlations strongly
suggest that systemic rotation may be intimately linked with the processes that
led to the formation of globular clusters and the stellar populations they
host.
