The Birds of North America online says that crows generally
avoid most physical contact, except in late winter and spring when
vocalizations and patrol flights occur. When they do fight it's
usually "over food and territorial trespass, 2 opponents jump-fly at
each other, belly to belly, pecking and clawing, or they stand near
each other and one tries to grasp the other by extending a foot;
both birds may lie on their sides trying to grab each other's feet,
or one bird stands on belly of other (Killiam 1989, Faucher 1994)."
Verbeek, N. A. and C. Caffrey (2002).
version 2.0. In The Birds of North America (A. F. Poole and F. B.
Gill, Editors). Cornell Lab of Ornithology, Ithaca, NY, USA.