  The techniques developed by Butscher in arXiv:math/0703469 for constructing
constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing
together spherical building blocks are generalized to handle less symmetric
initial configurations. The outcome is that the approximately CMC hypersurface
obtained by gluing the initial configuration together can be perturbed into an
exactly CMC hypersurface only when certain global geometric conditions are met.
These `balancing conditions' are analogous to those that must be satisfied in
the `classical' context of gluing constructions of CMC hypersurfaces in
Euclidean space, although they are more restrictive in the (n+1)-sphere case.
An example of an initial configuration is given which demonstrates this fact;
and another example of an initial configuration is given which possesses no
symmetries at all.
