The object of our investigation is the canonical operation of the automorphism group of a formally real fieldFonXF, the space of orderings ofF. For a naturally distinguished class of formally real fields, the so-called real-local fields, the Baer-Krull-bijection induces onXFthe structure of a module over the endomorphism ring of the group of archimedean classes ofF. We show that AutFacts onXFby affinities with respect to that module structure. Subsequently, this “arithmetization” of the operation is exemplarily applied to the question of transitivity (“When can any two orderings ofFbe transformed into each other by some automorphism ofF?"), and to the investigation of the subgroup of AutFgenerated by all order automorphism groups ofF.