We present a first model-independent calculation of ππ intermediate states in the hadronic-light-by-light (HLBL) contribution to the anomalous magnetic moment of the muon (g-2)_{μ} that goes beyond the scalar QED pion loop. To this end, we combine a recently developed dispersive description of the HLBL tensor with a partial-wave expansion and demonstrate that the known scalar-QED result is recovered after partial-wave resummation. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box a_{μ}^{π box}=-15.9(2)×10^{-11}. We then construct a suitable input for the γ^{*}γ^{*}→ππ helicity partial waves, based on a pion-pole left-hand cut and show that for the dominant charged-pion contribution, this representation is consistent with the two-loop chiral prediction and the COMPASS measurement for the pion polarizability. This allows us to reliably estimate S-wave rescattering effects to the full pion box and leads to our final estimate for the sum of these two contributions a_{μ}^{π box}+a_{μ,J=0}^{ππ,π-pole LHC}=-24(1)×10^{-11}.