In this paper we study the edge clique graph \(K(G)\) of many classes of intersection graphs \(G\) — such as graphs of boxicity \(\leq k\), chordal graphs and line graphs. We show that in each of these cases, the edge clique graph \(K(G)\) belongs to the same class as \(G\). Also, we show that if \(G\) is a \(W_4\)-free transitivity orientable graph, then \(K(G)\) is a weakly \( \theta \)-perfect graph.