  The doubly-polarized production of $W^\pm Z$ pairs at the Large Hadron
Collider (LHC) is presented at next-to-leading order (NLO) accuracy both for
the electroweak (EW) and QCD corrections, including a detailed description of
the calculational method using the double-pole approximation. Numerical results
at the 13 TeV LHC are presented in particular for the $W^- Z$ case in the
$e^-\bar{\nu}_e \mu^+\mu^-$ channel using ATLAS fiducial cuts and for polarized
distributions defined in the $WZ$ center-of-mass system. The NLO EW corrections
relative to the NLO QCD predictions are found to be smaller than $5\%$ in most
kinematic distributions, but can reach the level of $10\%$ in some
distributions such as lepton transverse momentum distributions or rapidity
separation between the electron and the $Z$ boson. EW corrections are not
uniform for different polarizations. A comparison between the new ATLAS
measurement of polarization fractions to our theoretical prediction is
presented.
