  J. Choe and M. Soret have constructed two families, which they call even and
odd, of closed embedded minimal surfaces in the round $3$-sphere, all of which
can be interpreted as desingularizations of certain unions of multiple Clifford
tori intersecting with high symmetry along two great circles. They construct
each odd surface by proving the existence of a minimal disc solving a Plateau
problem in a certain prism. Here we show these closed embedded minimal surfaces
are uniquely characterized by the boundary of their intersection with the
prism, and moreover the intersection is graphical with respect to rotations
along a suitable axis of the prism. This implies as a corollary that some of
the high-genus desingularizations of Clifford tori constructed elsewhere by the
authors are in fact congruent to odd surfaces of Choe and Soret of the same
genus.
