  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 $ZZH$ 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.
