G-symplectic general linear methods with zero parasitism growth factors are able to preserve quadratic invariants and symplectic structure in a similar way to canonical Runge-Kutta methods. To obtain order $p$ for a method $(A,U,B,V)$, it is necessary that $\xi$ exists such that $(E\xi)(t) = B(D\eta)(t) + V \xi(t),$ for all $t$ such that $|t|\le p$, where $\eta = A \eta D + U\xi.$ If the method is G-symplectic the order conditions are interrelated and can be reduced to a smaller set. The special case of order 4 methods with $V=\mbox{diag}(1,-1)$ will be considered in detail. Although there are only two order conditions, there are additional constraints on a possible starting method, represented by $\xi$. This should not be regarded as a serious handicap because, in a constant step size implementation, a complicated starting method does not represent a computational overhead.