Using the fact that the Matsumoto K_{0}-group can be defined as the K_{0}-group associated to a certain stabilized C*-algebra which is itself a flow invariant of the shift space, one sees that it carries an ordered structure which is also a flow invariant. It is well known in the theory of C*-algebras associated to shifts of finite type, the so called Cuntz-Krieger algebras, that in this case the order structure is degenerate and redundant. The main goal of the present paper is to perform a further analysis of our previous description of the Matsumoto K_{0}-group of substitutional shift spaces, leading to a complete description of the order it carries as well. In contrast to the case for shifts of finite type, we shall give an example proving that we hence arrive at a finer flow invariant than what the group, in itself, offers.