Given a discriminant $D<0$, $D\equiv 0, 1\bmod{4}$, there is a well-known bijection between primitive (positive definite) reduced forms of that discriminant and ideal classes in a particular order of an imaginary quadratic extension determined by $D$. In the group formed by the classes of reduced forms under direct composition, the squares form a subgroup, which constitutes a genus; the other genera are cosets of that subgroup.
My question is: is there anything useful to be said about the ideal classes in the ideal class group whose corresponding forms are in the same genus (coset) of the form class group?