We analyze the eigenproperties of the SO and (OO)-O-16-O-18 ground-state ((3)Sigma) molecules in congruent electric and magnetic fields. The energy levels are found to exhibit avoided crossings for states with the same projection M of the total angular momentum on the common axis of the fields. The avoided crossings are due to a first-order Stark effect which connects intersecting levels of aligned Zeeman states of opposite parity and makes them strongly oriented. We exemplify the enhancement of orientation and other generic features of the combined electric and magnetic dipole interaction in polar (3)Sigma molecules by evaluating the expectation values of the orientation and alignment cosines as a function of field-strength parameters for a selection of states with \M\less than or equal to 5. The molecular states created by the congruent fields can be used to state-select and focus molecules, to orient the molecular axis in studies of collision stereodynamics, to facilitate assignments of spectral transitions, and in molecular trapping. In particular, we found that mixed-isotope O-2 and O-17(2) are the only chalcogen molecules which are suitable for spatial trapping in a static trap, owing to their rotational ground state which correlates with a low-field seeking Zeeman state. (C) 2000 American Institute of Physics. [S0021-9606(00)00108-2].