Measurements of the strong-phase difference between D⁰ and D⁰<over bar> → π⁺π^-π⁺π^- are performed in bins of phase space. The study exploits a sample of quantum-correlated DD<over bar> mesons collected by the BESIII experiment in e⁺e^- collisions at a center-of-mass energy of 3.773 GeV, corresponding to an integrated luminosity of 2.93 fb^-1. Here, D denotes a neutral charm meson in a superposition of flavor eigenstates. The reported results are valuable for measurements of the CP-violating phase γ (also denoted Φ₃) in B^± → DK^±, D → π⁺π^-π⁺π^- decays, and the binning schemes are designed to provide good statistical sensitivity to this parameter. The expected uncertainty on y arising from the precision of the strong-phase measurements, when applied to very large samples of B-meson decays, is around 1.5° or 2°, depending on the binning scheme. The binned strong-phase parameters arc combined to give a value of F₊^4π = 0.746 ± 0.010 ± 0.004 for the CP-even fraction of D⁰ → π⁺π^-π⁺π^- decays, which is around 30% more precise than the previous best measurement of this quantity.