Using 448.0 x 10⁶ ψ(3686) events collected with the BESIII detector, an amplitude analysis is performed for ψ(3686)→γχ_{c1}, χ_{c1} → ηπ⁺π⁻ decays. The most dominant two-body structure observed is α₀(980)^± π^∓; α₀(980)^±→ηπ^±. The α₀(980) line shape is modeled using a dispersion relation, and a significant nonzero α₀(980) coupling to the η′π channel is measured. We observe χ_{c1} → α₂(1700)π production for the first time, with a significance larger than 17σ. The production of mesons with exotic quantum numbers, J^{PC} = 1⁻⁺, is investigated, and upper limits for the branching fractions χ_{c1}→π₁(1400)^±π^∓, χ_{c1} → π₁(1600)^±π^∓, and χ_{c1} → π₁(2015)^±π^∓, with subsequent π₁(X)^± → ηπ^± decay, are determined.