The scheduled launch of the LISA Mission in the next decade has called attention to the gravitational self-force problem. Despite an extensive body of theoretical work, long-time numerical computations of gravitational waves from extreme-massratio-inspirals remain challenging. This work proposes a class of numerical evolution schemes suitable to this problem based on Hermite integration. Their most important feature is time-reversal symmetry and unconditional stability, which enables these methods to preserve symplectic structure, energy, momentum and other Noether charges over long time periods. We apply Noether's theorem to the master fields of black hole perturbation theory on a hyperboloidal slice of Schwarzschild spacetime to show that there exist constants of evolution that numerical simulations must preserve. We demonstrate that time-symmetric integration schemes based on a 2-point Taylor expansion (such as Hermite integration) numerically conserve these quantities, unlike schemes based on a 1-point Taylor expansion (such as Runge-Kutta). This makes timesymmetric schemes ideal for long-time EMRI simulations.