Recent measurements of B0→ππ decays reveal two features that are in conflict with conventional calculations: the channel B0(B̄0)→π+π− shows a large direct CP-violating asymmetry, and the channel B0(B̄0)→π0π0 has an unexpectedly high branching ratio. We show that both features can be understood in terms of strong-interaction mixing of ππ and DD̄ channels in the isospin-zero state, an effect that is important because of the large experimentally observed ratio Γ(B0/B̄0→D+D−)/Γ(B0/B̄0→π+π−)≈50. Our dynamical model correlates the branching ratios and the CP-violating parameters C and S, for the decays B0(B̄0)→π+π−, B0(B̄0)→π0π0, B0(B̄0)→D+D− and B0(B̄0)→D0D̄0.