Inspired by a construction due to Hitchin [20], we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kähler metrics found by Gauduchon–Ornea [14]. We also show that the Inoue complex surfaces with b 2 = 0 do not admit bihermitian metrics. This completes the classification of the compact complex surfaces admitting strongly bihermitian metrics.