  Line-of-sight kinematic studies indicate that many Galactic globular clusters
have a significant degree of internal rotation. However, three-dimensional
kinematics from a combination of proper motions and line-of-sight velocities
are needed to unveil the role of angular momentum in the formation and
evolution of these old stellar systems. Here we present the first quantitative
study of internal rotation on the plane-of-the-sky for a large sample of
globular clusters using proper motions from Gaia DR2. We detect signatures of
rotation in the tangential component of proper motions for 11 out of 51
clusters at a $>$3-sigma confidence level, confirming the detection reported in
Gaia collaboration et al. (2018) for 8 clusters, and additionally identify 11
GCs with a 2-sigma rotation detection. For the clusters with a detected global
rotation, we construct the two-dimensional rotation maps and proper motion
rotation curves, and we assess the relevance of rotation with respect to random
motions ($V/\sigma\sim0.08-0.51$). We find evidence of a correlation between
the degree of internal rotation and relaxation time, highlighting the
importance of long-term dynamical evolution in shaping the clusters current
properties. This is a strong indication that angular momentum must have played
a fundamental role in the earliest phases of cluster formation. Finally,
exploiting the spatial information of the rotation maps and a comparison with
line-of-sight data, we provide an estimate of the inclination of the rotation
axis for a subset of 8 clusters. Our work demonstrates the potential of Gaia
data for internal kinematic studies of globular clusters and provides the first
step to reconstruct their intrinsic three-dimensional structure.
