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About a third of the way into A Taxonomy, by David Howe, the shaman and
hero are described in general under Complexity, Compound, Inclusive.
Following is the relevant section:
'An Inclusive Compound piece has the option, on any one move, of moving
as one or more of its component piece moves. Such pieces tend to be
extremely powerful and should be used with care.
eg. a Super Cardinal may move either as a Bishop or as a Knight, or may
move first as a Bishop followed by a Knight move (in the same direction),
or may move first as a Knight followed by a Bishop move (in the same
direction).' 

I can offer a piece that isn't in David's work though: a chesimal; a
'piece' made up of several chesspieces, generally of a few different
types, that all may move [1 at a time, legally] every turn, but must all
basically be in contact with [touching] each other at the beginning and end
of the move. This 'piece' can take hits; that is, it can lose some of its
units, and continue to exist, still moving all its remaining units. And it
can capture as many pieces as it can legally move onto in one turn. :-)

I suspect others, like Mats, for example, might offer other types of
pieces, but it may depend on just how broadly or narrowly you define
categories and subcategories. Enjoy.