  We study neutrino-induced charged-current (CC) $\pi^0$ production on carbon
nuclei using events with fully imaged final-state proton-$\pi^0$ systems. Novel
use of final-state correlations based on transverse kinematic imbalance enable
the first measurements of the struck nucleon's Fermi motion, of the
intranuclear momentum transfer (IMT) dynamics, and of the final-state hadronic
momentum configuration in neutrino pion production. Event distributions are
presented for i) the momenta of neutrino-struck neutrons below the Fermi
surface, ii) the direction of missing transverse momentum characterizing the
strength of IMT, and iii) proton-pion momentum imbalance with respect to the
lepton scattering plane. The observed Fermi motion and IMT strength are
compared to the previous MINERvA measurement of neutrino CC quasielastic-like
production. The measured shapes and absolute rates of these distributions, as
well as the cross-section asymmetries show tensions with predictions from
current neutrino generator models.
