Using 448.0×106 ψ(3686) events collected with the BESIII detector, an amplitude analysis is performed for ψ(3686)→γχc1, χc1→ηπ+π- decays. The most dominant two-body structure observed is a0(980)±π+; a0(980)±→ηπ±. The a0(980) line shape is modeled using a dispersion relation, and a significant nonzero a0(980) coupling to the η′π channel is measured. We observe χc1→a2(1700)π production for the first time, with a significance larger than 17σ. The production of mesons with exotic quantum numbers, JPC=1-+, is investigated, and upper limits for the branching fractions χc1→π1(1400)±π+, χc1→π1(1600)±π+, and χc1→π1(2015)±π+, with subsequent π1(X)±→ηπ± decay, are determined.