A global analysis of ATLAS and CMS measurements reveals that, at mid-rapidity, the directly-produced χ c 1 , χ c 2 and J/ ψ mesons have differential cross sections of seemingly identical shapes, when presented as a function of the mass-rescaled transverse momentum, p 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 T -differential cross sections. Remarkably, accurate kinematic cancellations among the various 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 χ c 1 and χ c 2 polarizations, the only observables not yet measured: they should be almost maximally different from one another, and from the J/ ψ polarization, a striking exception in the global panorama of quarkonium production. Measurements of the difference between the χ c 1 , χ c 2 and J/ ψ polarizations, complementing the observed identity of momentum dependences, represent a decisive probe of NRQCD.