Let n be a positive integer. A collection \(\mathcal{S}\) of subsets of \([n]=\{1,\ldots ,n\}\) is called symmetric if \(X\in \mathcal{S}\) implies \(X^*\in \mathcal{S}\) , where \(X^*:=\{i\in [n]:n-i+1\notin X\}\) . As the main results of this paper, one shows that in each of the three types of separation relations: strong, weak and chord ones, the following “purity phenomenon” takes place: all inclusion-wise maximal symmetric separated collections in \(2^{[n]}\) have the same cardinality. These give “symmetric versions” of well-known results on the purity of usual strongly, weakly and chord separated collections of subsets of [n], and in the case of weak separation, this extends a result due to Karpman on the purity of symmetric weakly separated collections in \(\left( {\begin{array}{c}[n]\\ n/2\end{array}}\right) \) for n even.
