We describe two approaches to classifying the possible monodromy cones C
arising from nilpotent orbits in Hodge theory. The first is based upon the
observation that C is contained in the open orbit of any interior point N in C
under an associated Levi subgroup determined by the limit mixed Hodge
structure. The possible relations between the interior of C its faces are
described in terms of signed Young diagrams.
The second approach is to understand the Tannakian category of nilpotent
orbits via a category D introduced by Deligne in a letter to Cattani and
Kaplan. In analogy with Hodge theory, there is a functor from D to a
subcategory of SL(2)-orbits. We prove that these fibers are, roughly speaking,
algebraic. We also give a correction to a result of K. Kato.