  This paper examines a systematic method to construct a pair of
(inter-related) root systems for arbitrary Coxeter groups from a class of
non-standard geometric representations. This method can be employed to
construct generalizations of root systems for a large family of groups
generated only by involutions. We then give a characterization of Coxeter
groups, among these groups, in terms of such paired root systems. Furthermore,
we use this method to construct and study the paired root systems for
reflection subgroups of Coxeter groups.
