  We present a study of the polarization observables of the $W$ and $Z$ bosons
in the process $p p \to W^\pm Z\to e^\pm \nu_e \mu^+\mu^-$ at the 13 TeV Large
Hadron Collider. The calculation is performed at next-to-leading order,
including the full QCD corrections as well as the electroweak corrections, the
latter being calculated in the double-pole approximation. The results are
presented in the helicity coordinate system adopted by ATLAS and for different
inclusive cuts on the di-muon invariant mass. We define left-right charge
asymmetries related to the polarization fractions between the $W^+ Z$ and $W^-
Z$ channels and we find that these asymmetries are large and sensitive to
higher-order effects. Similar findings are also presented for charge
asymmetries related to a P-even angular coefficient.
