In the paper we study the Bott–Chern and Aeppli cohomologies of the twistor space of a compact self-dual 4-manifold and we characterize the validity of the ∂∂¯-lemma. We also compute explicitly the Dolbeault cohomology of the twistor space Z of the flat 4-dimensional torus, which is known to not satisfy the ∂∂¯ lemma.
