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This is joint work with Ichihara and Ozawa We show that strongly irreducible Heegaard splittings can be organized into classes, by attaching separating then non-separating handles. With only separating handles, one always gets separating incompressible surfaces. Attaching the non-separating handles can give compressing disks for the surfaces in only a small number of cases, by a result of Y.Q. Wu. This gives some more evidence of why non Haken examples are difficult to find.