  The article considers symmetric general linear methods, a class of numerical
time integration methods which, like symmetric Runge--Kutta methods, are
applicable to general time--reversible differential equations, not just those
derived from separable second--order problems. A definition of time--reversal
symmetry is formulated for general linear methods, and criteria are found for
the methods to be free of linear parasitism. It is shown that symmetric
parasitism--free methods cannot be explicit, but a method of order $4$ is
constructed with only one implicit stage. Several characterizations of symmetry
are given, and connections are made with $G$--symplecticity. Symmetric methods
are shown to be of even order, a suitable symmetric starting method is
constructed and shown to be essentially unique. The underlying one--step method
is shown to be time--symmetric. Several symmetric methods of order $4$ are
constructed and implemented on test problems. The methods are efficient when
compared with Runge--Kutta methods of the same order, and invariants of the
motion are well--approximated over long time intervals.
