  We study the spin-exotic $J^{PC} = 1^{-+}$ amplitude in single-diffractive
dissociation of 190 GeV$/c$ pions into $\pi^-\pi^-\pi^+$ using a hydrogen
target and confirm the $\pi_1(1600) \to \rho(770) \pi$ amplitude, which
interferes with a nonresonant $1^{-+}$ amplitude. We demonstrate that
conflicting conclusions from previous studies on these amplitudes can be
attributed to different analysis models and different treatment of the
dependence of the amplitudes on the squared four-momentum transfer and we thus
reconcile their experimental findings. We study the nonresonant contributions
to the $\pi^-\pi^-\pi^+$ final state using pseudo-data generated on the basis
of a Deck model. Subjecting pseudo-data and real data to the same partial-wave
analysis, we find good agreement concerning the spectral shape and its
dependence on the squared four-momentum transfer for the $J^{PC} = 1^{-+}$
amplitude and also for amplitudes with other $J^{PC}$ quantum numbers. We
investigate for the first time the amplitude of the $\pi^-\pi^+$ subsystem with
$J^{PC} = 1^{--}$ in the $3\pi$ amplitude with $J^{PC} = 1^{-+}$ employing the
novel freed-isobar analysis scheme. We reveal this $\pi^-\pi^+$ amplitude to be
dominated by the $\rho(770)$ for both the $\pi_1(1600)$ and the nonresonant
contribution. We determine the $\rho(770)$ resonance parameters within the
three-pion final state. These findings largely confirm the underlying
assumptions for the isobar model used in all previous partial-wave analyses
addressing the $J^{PC} = 1^{-+}$ amplitude.
