  In many extensions of the standard model, there exist a few extra Higgs
bosons. Suppose a heavy neutral Higgs boson H is discovered at the LHC, one
could then investigate CP and CPT~ properties of its couplings to a pair of $Z$
bosons through $H \to ZZ \to 4\ell$. We use the helicity-amplitude method to
write down the most general form for the angular distributions of the four
final-state leptons, which can cover the case of CP-even, -odd, and -mixed
state for the Higgs boson. We figure out there are 9 types of angular
observables and all the $H$ couplings to $Z$ bosons can be fully determined by
exploiting them. A Higgs-boson mass of 260 GeV below the $t\bar t$ threshold is
illustrated with full details. With a total of $10^3$ events of $H \to ZZ \to
4\ell$, one can determine the couplings up to 12-20\% uncertainties.
