  We consider the Jacobian Kummer surface $X$ of a genus two curve $C$. We
prove that the Hutchinson-Weber involution on $X$ degenerates if and only if
the Jacobian $J(C)$ is Comessatti. Also we give several conditions equivalent
to this, which include the classical theorem of Humbert. The key notion is the
Weber hexad. We include explanation of them and discuss the dependence between
the conditions of main theorem for various Weber hexads. It results in "the
equivalence as dual six". We also give a detailed description of relevant
moduli spaces. As an application, we give a conceptual proof of the computation
of the patching subgroup for generic Hutchinson-Weber involutions.
