  In our work we investigate quotient structures and quotient spaces of a space
of orderings arising from subgroups of index two. We provide necessary and
sufficient conditions for a quotient structure to be a quotient space that,
among other things, depend on the stability index of the given space. The case
of the space of orderings of the field Q(x) is particularly interesting, since
then the theory developed simplifies significantly. A part of the theory
firstly developed for quotients of index 2 generalizes in an elegant way to
quotients of index 2^n for arbitrary finite n. Numerous examples are provided.
