We calculate the cross section of inclusive direct J/ψ hadroproduction at next-to-leading order within the factorization formalism of nonrelativistic quantum chromodynamics, including the full relativistic corrections due to the intermediate 1S(0)[8], 3S(1)[8], and 3P(J)[8] color-octet states. We perform a combined fit of the color-octet long-distance matrix elements to the transverse-momentum (p(T)) distributions measured by CDF at the Fermilab Tevatron and H1 at DESY HERA and demonstrate that they also successfully describe the p(T) distributions from PHENIX at BNL RHIC and CMS at the CERN LHC as well as the photon-proton c.m. energy and (with worse agreement) the inelasticity distributions from H1. This provides a first rigorous test of nonrelativistic QCD factorization at next-to-leading order. In all experiments, the color-octet processes are shown to be indispensable.