  A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors
have the same parity of number of circuits. In \cite{ADJLS} we proved that the
only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite
graph of girth 4 is $K_{3,3}$, and conjectured \cite[Conjecture 3.6]{ADJLS}
that the only essentially 4--edge-connected cubic bipartite graphs are
$K_{3,3}$, the Heawood graph and the Pappus graph.
  There exists a characterization of symmetric configurations $n_3$ %{\bf
decide notation and how to use it in the rest of the paper} due to Martinetti
(1886) in which all symmetric configurations $n_3$ can be obtained from an
infinite set of so called {\em irreducible} configurations \cite{VM}. The list
of irreducible configurations has been completed by Boben \cite{B} in terms of
their {\em irreducible Levi graphs}.
  In this paper we characterize irreducible pseudo 2--factor isomorphic cubic
bipartite graphs proving that the only pseudo 2--factor isomorphic irreducible
Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained
characterization allows us to partially prove the above Conjecture.
