Symplectic or symmetric integration methods do not only preserve geometrical structures of the flow of the differential equation, but they have also favourable properties concerning their global error when the integration is performed over a very long time. The subject of this paper is to provide new insight into this phenomenon. For problems with periodic solution and for integrable systems we prove that the error growth is only linear for symplectic and symmetric methods, compared to a quadratic error growth in the general case. Furthermore, for symmetric collocation methods we explain a variable-stepsize implementation which does not destroy the abovementioned properties.