A new fourth order predictor–corrector integration scheme is presented. The unique feature of the new algorithm and what distinguishes it from a Gear predictor–corrector is that the method is derived from the Trotter decomposition of a specially formulated evolution operator and as such, is both symplectic and reversible. In addition, the method retains the useful property of Gear methods that only one force evaluation per time step is required. The new integrator is tested on a harmonic plus quartic oscillator and the Henon–Heiles system. Comparisons are made to the second order velocity Verlet integrator, the true fourth order Yoshida/Suzuki schemes and fourth order Gear. In all cases, the new method works well, giving energy conservation and trajectories of much better quality than velocity Verlet and of comparable quality to the results of the true fourth order schemes for the same computational cost as velocity Verlet.