This paper focuses on fixed-time synchronization (FTS) and preassigned-time synchronization (PTS) of bidirectional associative memory memristive neural networks (BAMMNNs) with mixed-time delays via event-triggered control (ETC). Firstly, by using Lyapunov stability theory, fixed/preassigned-time stability lemmas and inequality techniques, results on FTS and PTS of BAMMNNs are derived. Secondly, compared to asymptotic synchronization and finite-time synchronization, the FTS and PTS studied here achieve faster convergence speeds and more precise settling times. Thirdly, the model incorporates state switching, time-varying and distributed delays; specifically, the time delays do not require differentiability, which enhances the generality of the results. Additionally, a segmented ETC strategy is designed to suit the dual-layer structure of BAMMNNs, where control actions are executed based on set triggering conditions, thus significantly reducing information transmission power consumption. Finally, a numerical simulation example is provided to verify the correctness of the results.