We describe numerical simulations of the miscible Rayleigh-Taylor (RT) instability driven by a complex acceleration history, g(t), with initially destabilizing acceleration, g > 0, an intermediate stage of stabilizing deceleration, g < 0, and subsequent destabilizing acceleration, g > 0. Initial perturbations with both single wavenumber and a spectrum of wavenumbers (leading to a turbulent front) have been considered with these acceleration histories. We find in the single-mode case that the instability undergoes a so-called phase inversion during the first acceleration reversal from g > 0 to g < 0. If the zero-crossing of g(t) occurs once the instability growth has reached a state of nonlinear saturation, then hitherto rising bubbles and falling spikes reverse direction and collide, causing small-scale structures to emerge and enhancing molecular mixing in the interfacial region. Beyond the second stationary point of g(t) where once again g > 0, the horizontal mean density profile becomes RT-unstable and the interfacial region continues to enlarge. Secondary Kelvin-Helmholtz-unstable structures on the near-vertical sheared edges of the primary bubble have an Atwood-number-dependent influence on the primary RT growth rate. This Atwood number dependence appears to occur because secondary instabilities strongly promote mixing, but the formation of these secondary structures is suppressed at large density differences. For multi-mode initial perturbations, we have selected an initial interfacial amplitude distribution h0 (λ) that rapidly achieves a self-similar state during the initial g > 0 acceleration. The transition from g > 0 to g < 0 induces significant changes in the flow structure. As with the single-mode case, bubbles and spikes collide during phase inversion, though in this case the interfacial region is turbulent, and the region as a whole undergoes a period of enhanced structural breakdown. This is accompanied by a rapid increase in the rate of molecular mixing, and increasing isotropy within the region. During the final stage of g > 0 acceleration, self-similar RT mixing re-emerges, together with a return to anisotropy. We track several turbulent statistical quantities through this complex evolution, which we present as a resource for the validation and refinement of turbulent mix models.