This paper proves a uniqueness result for 2-spheres that split a knotted handlebody in the
3-sphere along three parallel disks. We apply the result to study the symmetry of knotted
handlebodies, measured by the mapping class group. In particular, the chirality of 610 in the
handlebody-knot table, which was previously unknown, is determined. An infinite family of
hyperbolic handlebody-knots with homeomorphic exteriors is also constructed.
