  It is well known that a constant O(n,n,Z) transformation can relate different
string backgrounds with n commuting isometries that have very different
geometric and topological properties. Here we construct discrete families of
(flux) backgrounds on internal manifolds of different topologies by performing
certain coordinate dependent O(d,d) transformations, where d is the dimension
of the internal manifold. Our two principal examples include respectively the
family of type IIB compactifications with D5 branes and O5 planes on
six-dimensional nilmanifolds, and the heterotic torsional backgrounds.
