Summary: Unlike graphs, digraphs or binary relational structures, the 2-structures do not define precise links between vertices. They only yield an equivalence between links. Also 2-structures provide a suitable generalization in the framework of clan decomposition. Let \(\sigma\) be a 2-structure. A subset \(C\) of \(V(\sigma)\) is a clan of \(\sigma\) if for each \(v\in V(\sigma)\setminus C\), \(v\) is linked in the same way to all the elements of \(C\). A 2-structure \(\sigma\) is clan primitive if \(|V(\sigma)|\geq 3\) and all its clans are trivial. A clan primitive 2-structure \(\sigma\) is critically primitive if \(\sigma[V(\sigma)\setminus\{v\}]\) is not primitive for every \(v\in V(\sigma)\). A clan of cardinality 2 is a twin pair. A 2-structure \(\sigma\) is twin primitive if \(|V(\sigma)|\geq 3\) and \(\sigma\) has no twin pairs. A twin primitive 2-structure \(\sigma\) is critically twin primitive if \(\sigma [V(\sigma)\setminus\{v\}]\) is not twin primitive for every \(v\in V(\sigma)\). First, a twin decomposition is provided for 2-structures. Second, twin primitivity is studied by proving analogues of results on clan primitivity. Third, the notion of critically closed subset is introduced to characterize critically twin primitive 2-structures.