  We discuss special type of polarization asymmetry (``handedness") in the
process $e^+e^-\to 4\pi$, when one of initial particles is longitudinally
polarized. The asymmetry is proportional to the degree of polarization and to
the width to mass ratio of rho-meson. It can reach 4-5\% in some kinematical
region. Both channels $2 \pi^+ 2 \pi^-$ and $2 \pi^0 \pi^+ \pi^-$ are
considered in the framework of the effective chiral lagrangian with vector
mesons, in the energy range $\sim 1$ GeV. The corresponding total cross
sections are also calculated.
