A hypergraph is selfdual if it is isomorphic to its dual. Another way of expressing this is to say that the vertexhyperedge incidence matrix can be made symmetric by permutating the columns. Thus these graphs in questions are just the rowintersection graphs of symmetric (01) matrices, and therefore special competition graphs.
In the second step of the dualityapproach of recognizing intersection graphs of selfdual hypergraphs, one needs a method to recognize selfdual hypergraphs. ....
Now the Krausztype characterization obviously reads as follows:
The following statements are equivalent for every graph G=(V,E):

Selfdual hypergraphs may have different representations as symmetric matrices, i.e. there may be different isomorphisms between the hypergraph and its dual.. But for such matrices, we have two extremes: Selfdual hypergraphs where every symmetrical vertexhyperedge incidence matrix has no `1´ in the main diagonal, and those with no `0´. Selfdual hypergraphs of the first kind are just the neighborhood hypergraphs of graphs, those of the second kind are just the closedneighborhood hypergraphs of graphs. But the intersection graph of the neighborhood hypergraph of a graph G is just the 2step graph of G, whereas the intersection graph of the closedneighborhood hypergraph of G equals the square of G. Therefore, the above theorem yields the following Krausztype characterizations of squares and 2step graphs:
[M67]: A graph G=(V,E) is the square of some graph if and only if there is some covering of the its edges by complete subgraphs S_{x}, xV, such that each S_{x} contains x, and xS_{y} if and only if yS_{x}. 
[AV73]: A graph G=(V,E) is the 2step graph of some graph if and only if there is some covering of the its edges by complete subgraphs S_{x}, xV, such that no S_{x} contains x, and xS_{y} if and only if yS_{x}, for distinct x,y . 
These characterizations are not too usefulactually recognizing squares of graphs is NPcomplete [MS94].