A permutation [sigma] of [1, n] is a self-complementing permutation of a self-complementary graph of order n if and only if all the orbits of [sigma] have their cardinalities congruent to 0 (mod 4) except, possibly, one orbit of cardinality 1.
The 2-uniform self-complementary hypergraphs are exactly self-complementary graphs. This class of graphs has been independently discovered by Ringel and Sachs who proved the following.
Sciriha, "The main eigenvalues and number of walks in
self-complementary graphs," Linear and Multilinear Algebra, 2013.