Include all examples of such theories.
Motivic cohomology and Motivic homology
Deligne cohomology and Deligne homology, see also Deligne-Beilinson cohomology, Absolute Hodge cohomology
Morphic cohomology and Lawson homology, see also Topological cycle cohomology
General notions: Bloch-Ogus cohomology, Gillet cohomology, Cohomology with compact supports
nLab page on D40 More about Bloch-Ogus theories and Beilinson theories