As it has been proved, the use of numerical propagators with some properties of symmetry, like symplectic and symmetric linear multi-step methods, is a good practice in the numerical propagation of orbital problems. In the other hand, multirevolution methods are good candidates for the long-time propagation of quasiperiodic problems. In this paper, we construct multi-revolution methods with some properties of symmetry, based on linear multi-step methods, and with arbitrary order of convergence. We study some of its properties of order. We also perform some comparison with the classical multi-revolution methods for several well-known problems, the perturbed harmonic oscillator and the perturbed Kepler problem formulated in such variables that the equations of motion are regularized and linearized what gives a smaller increment of the errors.