  (Abridged) We consider the polarization properties of optically thin
synchrotron radiation emitted by relativistically moving electron--positron
jets carrying large-scale helical magnetic fields. In our model, the jet is
cylindrical, and the emitting plasma moves parallel to the jet axis with a
characteristic Lorentz factor $\Gamma$. We draw attention to the strong
influence that the bulk relativistic motion of the emitting relativistic
particles has on the observed polarization.
  We conclude that large-scale magnetic fields can explain the salient
polarization properties of parsec-scale AGN jets. Since the typical degrees of
polarization are $\leq 15%$, the emitting parts of the jets must have
comparable rest-frame toroidal and poloidal fields. In this case, most
relativistic jets are strongly dominated by the toroidal magnetic field
component in the observer's frame, $B_\phi/B_z \sim \Gamma$. We also discuss
the possibility that relativistic AGN jets may be electromagnetically (Poynting
flux) dominated. In this case, dissipation of the toroidal magnetic field (and
not fluid shocks) may be responsible for particle acceleration.
