  We find a geometric invariant of isotopy classes of strongly irreducible
Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a
theorem of R Weidmann, proved here in the appendix, we show that a closed,
totally orientable Seifert fibered space M has infinitely many isotopy classes
of Heegaard splittings of the same genus if and only if M has an irreducible,
horizontal Heegaard splitting, has a base orbifold of positive genus, and is
not a circle bundle. This characterizes precisely which Seifert fibered spaces
satisfy the converse of Waldhausen's conjecture.
