The biaxial nematic phase is generally taken, either explicitly or implicitly, to have D(2h) point group symmetry. However, it is possible for the biaxial phase to have a lower symmetry depending on that of its constituent molecules. Here we develop a molecular field theory for a nematogen composed of C(2h) molecules in terms of the nine independent second rank orientational order parameters defining the C(2h) biaxial nematic. In addition, there is a rank one order parameter constructed from two pseudovectors which is only nonzero in the C(2h) phase. The theory is simplified by removing all but the three dominant order parameters. The predicted phase behavior is found to be rich with three possible biaxial nematic phases and with the transitions involving a biaxial nematic phase exhibiting tricritical points.