The chief justice gets to shape coalitions through the assignments of the cases.
Constitutional law is a tricky business and the justices often will influence one another over one legal nuance or another. Moreover, though each justice of course hears and decides on all the cases misses an essential point: the human element, the personal chemistry among the justices. It is really no different than who gets to sponsor a bill in the Legislature; that person, and the approach he/she takes, will obviously influence others.
I'm sorry for being so dense -- but I still don't have a clear idea of what "assignment of cases" means in the concrete. :-(