  We define a notion of Hempel distance for one-sided Heegaard splittings and
show that the existence of alternate surfaces restricts distance for one-sided
splittings in a manner similar to Hartshorn's and Scharlemann-Tomova's results
for two-sided splittings. We also show that every geometrically compressible
one-sided Heegaard surface in a non-Haken 3-manifold is stabilized, and show
that the mapping class group of the two-sided Heegaard splitting induced by a
high distance one-sided splitting is isomorphic to the fundamental group of the
one-sided surface.
