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Based on a general form of the effective vertex functions for the decays ofP-wave charmonia $\chicj$, angular distribution formulas for the subsequentdecays $\chicj\rightarrow \jpsi \gamma$ and $\jpsi \to \mu^+\mu^-$ are derived.The formulas are the same as those obtained in a different approach in theliterature. Our formulas are expressed in a more general form, including parityviolation effects and the full angular dependence of $\jpsi$ and muon in thecascade decay $\chicj\to\jpsi\gamma\to\mu^+\mu^-\gamma$. The $\chicj$polarization observables are expressed in terms of rational functions of thespin density matrix elements of $\chicj$ production. Generalizedrotation-invariant relations for arbitrary integer-spin particles are alsoderived and their expressions in terms of observable angular distributionparameters are given in the $\chi_{c1}$ and $\chi_{c2}$. To complement ourprevious direct-$\jpsi$ polarization result, we also discuss the impact on theobservable prompt-$\jpsi$ polarization. As an illustrative application of ourangular distribution formulas, we present the angular distributions in terms ofthe tree-level spin density matrix elements of $\chi_{c1}$ and $\chi_{c2}$production in several different frames at the Large Hadron Collider. Moreover,a reweighting method is also proposed to determine the entire set of theproduction spin density matrix elements of the $\chi_{c2}$, some of whichdisappear or are suppressed for vanishing higher-order multipole effects makingthe complete extraction difficult experimentally.

Spin correlations in polarizations of P-wave charmonia $\chi_{cJ}$ and impact on $J/\psi$ polarization