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We compute the ${\mathcal O}(α_s^2)$ corrections to the exclusive channels $e^+e^-\to J/ψ+χ_{cJ}$ ($J=0,1,2$) at $\sqrt{s}=10.58$ GeV within the nonrelativistic QCD (NRQCD) factorization framework. The validity of NRQCD factorization at ${\cal O}(α_s^2)$ has been confirmed for these double-charmonium exclusive production processes. We analyze the impact of the $\mathcal{O}(α_s^2)$ corrections on the polarized and unpolarized cross sections, as well as the $J/ψ$ angular distributions, which largely reduce the renormalization scale dependence but increase the ${\mathcal O}(α_s)$ NRQCD predictions to some extent for $χ_{c0,1}$. With high numerical accuracy, our predictions for $σ(J/ψ+χ_{c1,2})$ through ${\mathcal O}(α_s^2)$ are compatible with the upper limit of the \texttt{Belle} measurement. Although the theoretical prediction for $σ(J/ψ+χ_{c0})$ is consistent with both \texttt{Belle} and \texttt{BaBar} measurements within uncertainties, there still exists serious tension between the predicted and the measured profiles for the $J/ψ$ angular distribution. Regarding the bright observation prospect of the $e^+e^-\to J/ψ+χ_{c1,2}$ channels in \texttt{Belle 2} experiment, it is interesting to compare the future measurements with our NRQCD predictions. The more accurate measurement of $e^+e^-\to J/ψ+χ_{0}$ at \texttt{Belle 2} will also help to clarify the long-standing puzzle of $J/ψ$ angular distribution.