  Using data from the FOCUS (E831) experiment at Fermilab, we present new
measurements for the Cabibbo-suppressed decay mode $D^0 \to
\pi^-\pi^+\pi^-\pi^+$. We measure the branching ratio $\Gamma(D^0 \to\pi^+\pi^-
\pi^+\pi^-)/\Gamma(D^0 \to K^-\pi^+\pi^-\pi^+) = 0.0914 \pm 0.0018 \pm 0.0022$.
An amplitude analysis has been performed, a first for this channel, in order to
determine the resonant substructure of this decay mode. The dominant component
is the decay $D^0 \to a_1(1260)^+ \pi^-$, accounting for 60% of the decay rate.
The second most dominant contribution comes from the decay $D^0 \to
\rho(770)^0\rho(770)^0$, with a fraction of 25%. We also study the $a_1(1260)$
line shape and resonant substructure. Using the helicity formalism for the
angular distribution of the decay $D^0 \to \rho(770)^0\rho(770)^0$, we measure
a longitudinal polarization of $P_L = (71 \pm 4\pm 2)$%.
