How a strongly irreducible Heegaard splitting intersects a handlebody

In [Topology Appl. 90 (1998) 135] Scharleman showed that a strongly irreducible Heegaard splitting surface Q of a 3-manifold M can, under reasonable side conditions, intersect a ball or a solid torus in M in only a few possible ways. Here we extend those results to describe how Q can intersect a handlebody in M.