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

From identical S- and P-wave pT/M spectra to maximally distinct polarizations: probing NRQCD with chi states