  We present a new and improved model independent amplitude analysis of
reactions $\pi^+ n_\uparrow \to \pi^+ \pi^- p$ at 5.98 and 11.85 GeV/c and
$\pi^- p_\uparrow \to \pi^- \pi^+ n$ at 17.2 GeV/c measured with transversely
polarized targets at the CERN Proton Synchrotron. A clear signal for
$\sigma(750)$ resonance emerges in all four solutions for $S$-wave intensity
$I_S$ at 5.98 and 11.85 GeV/c and in both solutions for $S$-wave amplitude
$|\overline S|^2\Sigma$ at 17.2 GeV/c. Its $\pi^+\pi^-$ decay width is
estimated to be in the range 200--300 MeV. We find significant suppression of
$\rho^0$ production in the amplitudes $|U|^2\Sigma$, $|\overline N |^2\Sigma$
and, at 17.2 GeV/c, in $|L|^2\Sigma$. The mass dependence of amplitudes
$|\overline L |^2\Sigma$ and $|L|^2\Sigma$ shows unexpected structures within
the $\rho^0$ mass region which correlate the mass spectra corresponding to
opposite nucleon spins. These features of $P$-wave moduli reveal the essential
role of nucleon spin in pion production process and contradict the
factorization hypothesis. Our results emphasize the need for a systematic study
of pion production on the level of amplitudes in a new generation of dedicated
experiments with spin at the recently proposed high-intensity hadron
facilities.
