FWIW I was able to kinda sorta do that for basic recursion combinators: http://joypy.osdn.io/notebooks/Recursion_Combinators.html#co...

In "Functional Programming with Bananas, Lenses, & Barbed Wire" http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1... they talk about morphisms: Hylomorphism, Anamorphism, Catamorphism, Paramorphism.

If you break them down using Joy (programming language) in terms of the genrec general recursion combinator, they form simple patterns, which then suggest at least two other unnamed combinators:

Hylo-, Ana-, Cata-

    H == [P  ] [pop c ] [G          ] [dip F        ] genrec
    A == [P  ] [pop []] [G          ] [dip swap cons] genrec
    C == [not] [pop c ] [uncons swap] [dip F        ] genrec

Para-, ?-, ?-

    P == c  swap [P  ] [pop] [[F        ] dupdip G          ] primrec
    ? == [] swap [P  ] [pop] [[swap cons] dupdip G          ] primrec
    ? == c  swap [not] [pop] [[F        ] dupdip uncons swap] primrec

Etc.