The code:

Solution of W kinematics
Solid ellipses - W decay kinematics in LAB
Dashed circles - W decay kinematics in W CMS, radii defined by m_W/2=45 GeV.
Dotted horizontal line - Lorentz transformation LAB <--> CMS for electron
Dashed vectors show CMS momentum of electron - In general there are 2 CMS solutions for a single  electron vector in LAB. Neutrino flies in the opposite direction to electron in CMS frame.

Measured: electron momentum P_e^LAB (red), sign of recoil pZ (above plot)

Fig 2a. One solution of W kinematic

Location of centers  of ellipses are uniquely defined by the  momentum vector of reco electron (P_e^LAB) , what locks  2 possible values of W momentum (P_W), one positive and one negative. Those 2 solution are correlated with the beam which gave  high pT quark, the opposite beam gave see anti-quark.  The sign of the recoil pZ selects one of 2 solutions. By depolarizing either of the beams we can depolarize either quark or antiquark (with high probability of not mixing it up).

Fig 2b. Two solution of W kinematic