I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category has attached to it a multiset of source/targets, and each pair of morphisms can be composed along any common submultiset of source/targets.

Does anyone know of a reference in the literature / a page on the nLab where generalized notions of categories with "unoriented compositions," e.g. the above notion, are defined and named?

Most formalisms I know for dealing with this sort of thing start with something which is "oriented" in that it distinguishes sources and targets and equip it with extra structure allowing one to turn sources into targets and vice versa. But the example I have in mind doesn't have this property: the "morphisms" are finite graphs together with labels on some of their vertices such that no pair of labeled vertices is connected by an edge, and composition is giving by gluing together graphs along labeled vertices with shared labels. I can't interpret this example as, say, a symmetric monoidal category with duals because there are no cups or caps.