  It was shown by Bonahon-Otal and Hodgson-Rubinstein that any two genus-one
Heegaard splittings of the same 3-manifold (typically a lens space) are
isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang
that certain Seifert manifolds have distinct genus-two Heegaard splittings. In
an earlier paper, we presented a technique for comparing Heegaard splittings of
the same manifold and, using this technique, derived the uniqueness theorem for
lens space splittings as a simple corollary. Here we use a similar technique to
examine, in general, ways in which two non-isotopic genus-two Heegard
splittings of the same 3-manifold compare, with a particular focus on how the
corresponding hyperelliptic involutions are related.
