  Polarization measurements are usually considered as the most difficult
challenge for the QCD description of quarkonium production. In fact, global
data fits for the determination of the non-perturbative parameters of
bound-state formation traditionally exclude polarization observables and use
them as a posteriori verifications of the predictions, with perplexing results.
With a change of perspective, we move polarization data to the centre of the
study, advocating that they actually provide the strongest fundamental
indications about the production mechanisms, even before we explicitly consider
perturbative calculations.
  Considering psi(2S) and Y(3S) measurements from LHC experiments and
state-of-the-art NLO short-distance calculations in the framework of
non-relativistic QCD factorization (NRQCD), we perform a search for a kinematic
domain where the polarizations can be correctly reproduced together with the
cross sections, by systematically scanning the phase space and accurately
treating the experimental uncertainties. This strategy provides a
straightforward solution to the "quarkonium polarization puzzle" and reassuring
signs that the theoretical framework is reliable. At the same time, the results
expose unexpected hierarchies in the non-perturbative NRQCD parameters, that
open new paths towards the understanding of bound-state formation in QCD.
