  The first author introduced the circuit-cocircuit reversal system of an
oriented matroid, and showed that when the underlying matroid is regular, the
cardinalities of such system and its variations are equal to special
evaluations of the Tutte polynomial (e.g., the total number of
circuit-cocircuit reversal classes equals $t(M;1,1)$, the number of bases of
the matroid). By relating these classes to activity classes studied by the
first author and Las Vergnas, we give an alternative proof of the above results
and a proof of the converse statements that these equalities fail whenever the
underlying matroid is not regular. Hence we extend the above results to an
equivalence of matroidal properties, thereby giving a new characterization of
regular matroids.
