One general thing to keep in mind - just because two electronic states are of different symmetry does not mean they cannot couple or interact, because vibrational and rotational states are also involved. Two electronic states may have different symmetries but they can couple through vibrations - the overall symmetry of a state is a product of the symmetries of its respective electronic and vibrational counterparts.
E.g., benzene may be nominally D2h symmetry - with electronic states having symmetries derived from the D2h character table - but the nuclei aren't stationary. During a vibration, the symmetry changes. If there is a certain vibration that can cause the ground and electronic states to temporarily have the same overall symmetry classification, coupling of the states becomes possible. Vibronic coupling is one reason why spectroscopic transitions can often occur between electronic states that have seemingly incompatible symmetries - the transition occurs through a vibrational mode the satisfies the symmetry requirement for the selection rule. Similar vibronic effects can also have an effect on whether avoided crossings and noncrossing impacts excited-state relaxation processes.