  Based on a sample corresponding to 4.3 million produced tau-pair events, we
have studied hadronic dynamics in the decay tau- --> nu_tau pi- pi0 pi0 in data
recorded by the CLEO II detector operating at the CESR e+e- collider. The decay
is dominated by the process tau --> nu_tau a_1(1260), with the a_1 meson
decaying to three pions predominantly via the lowest dimensional (mainly
S-wave) a_1 --> rho pi Born amplitude. From fits to the Dalitz plot and angular
observables, we find significant additional contributions from amplitudes for
a_1 decay to sigma pi, f_0(1370) pi and f_2(1270) pi, as well as higher
dimensional a_1 --> rho pi and rho' pi amplitudes. The squared sigma pi
amplitude accounts for ~15% of the total tau- --> nu_tau pi- pi0 pi0 rate in
the models considered. We have searched for additional contributions from tau
--> nu_tau pi'(1300). We place 90% confidence level upper limits on the
branching fraction for this channel of between 1.0*10^{-4} and 1.9*10^{-4},
depending on the pi' decay mode. The pi- pi0 pi0 mass spectrum is parametrized
by a Breit-Wigner form with a mass-dependent width which is specified according
to the results of the Dalitz plot fits plus a coupling to an a_1 --> K* K
amplitude. From a chi^2 fit, we extract the pole mass and width of the a_1, as
well as the magnitude of the K* K coupling. We have also investigated the
impact of a possible contribution from the a_1'(1700) meson on this spectrum.
Finally, exploiting the parity-violating angular asymmetry in a_1 --> 3pi
decay, we determine the signed value of the tau neutrino helicity to be
h_{\nu_\tau} = -1.02 +- 0.13(stat.) +- 0.03(syst.+model), confirming the
left-handedness of the tau neutrino.
