This paper studies the relation between associativity of uninorms and geometry of their level sets which is enabled by adopting the concepts of web geometry, a branch of differential geometry, and the Reidemeister closure condition. Based on this result, the structure of some special classes of uninorms is described. Namely, it is the class of uninorms with involutive underlying t-norms and t-conorms and the class of uninorms with involutive underlying t-norms and idempotent underlying t-conorm (as well as the corresponding dual cases).