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

On the Number of Circuit-cocircuit Reversal Classes of an Oriented Matroid