This paper is concerned with differential equations of the form \(\dot x = X = A + B\). The assumption that the vector fields \(A\) and \(B\) can be integrated exactly enables one to integrate \(X\) by composition of the flows \(A\) and \(B\). Various symmetric compositions are investigated for order, complexity, and reversibility. Free Lie algebra theory leads to simple formulas for the number of determining equations for a method to have a particular order. The author then obtains a new, more accurate way of applying the resulting methods to compositions of an arbitrary first- order integrator, and their implementation and numerical performance is described in detail, using as illustrations separable and non-separable Hamiltonians.