  A global analysis of ATLAS and CMS measurements reveals that, at
mid-rapidity, the directly-produced $\chi_{c1}$, $\chi_{c2}$ and J/$\psi$
mesons have differential cross sections of seemingly identical shapes, when
presented as a function of the mass-rescaled transverse momentum, $p_{\rm
T}/M$. This identity of kinematic behaviours among S- and P-wave quarkonia is
certainly not a natural expectation of non-relativistic QCD (NRQCD), where each
quarkonium state is supposed to reflect a specific family of elementary
production processes, of significantly different $p_{\rm T}$-differential cross
sections. Remarkably, accurate kinematic cancellations among the variegated
NRQCD terms (colour singlets and octets) of its factorization expansion can
lead to a surprisingly good description of the data. This peculiar tuning of
the NRQCD mixtures leads to a clear prediction regarding the $\chi_{c1}$ and
$\chi_{c2}$ polarizations, the only observables not yet measured: they should
be almost maximally different from one another, and from the J/$\psi$
polarization, a striking exception in the global panorama of quarkonium
production. Measurements of the difference between the $\chi_{c1}$, $\chi_{c2}$
and J/$\psi$ polarizations, complementing the observed identity of momentum
dependences, represent a decisive probe of NRQCD.
