  The production of a Higgs boson in association with a $Z$ at an
electron-positron collider is one of the cleanest methods for the measurement
of the couplings of the Higgs boson. In view of the large production cross
section at energies a little above the threshold, it seems feasible to make a
more detailed study of the process by measuring the cross sections for
polarized $Z$ in order to measure possible anomalous $ZZH$ couplings. We show
that certain combinations of cross sections in $e^+e^- \to ZH$ with different
$Z$ polarizations help to enhance or isolate the effect of one of the two kinds
of anomalous $ZZH$ couplings possible on general grounds of CP and Lorentz
invariance. These combinations can be useful to get information on the $ZZH$
coupling in the specific contexts of an effective field theory,
two-Higgs-doublet models, and composite Higgs models, in a relatively
model-independent fashion. We find in particular that the longitudinal helicity
fraction of the $Z$ is expected to be insensitive to anomalous couplings, and
would be close to its value in the standard model in the scenarios we consider.
We also discuss the sensitivity of the proposed measurements to the anomalous
couplings, including longitudinal beam polarizations, which suppress
backgrounds, and can improve the sensitivity if appropriately chosen.
