In reexamining chapter 11 of Jackson's Classical Electrodynamics, especially equations 11.148, it seems obvious that in placing the E and B transformation values into the electro-magnetic field-strength tensor one is ignoring the standard rules which do not allow combining polar vectors and axial vectors, or in this case, scalars and pseudoscalars. The result is that one loses parity information. That usage apparently originated with Minkowski. How did he motivate that apparent lapse of rigor? How did the receivers of that usage justify its acceptance? On page 558 Jackson states "Transformation (11.149) shows that E and B have no independent existence. A purely electric or magnetic field in one coordinate system will appear as a mixture of electric and magnetic fields in another coordinate frame." Isn't he ignoring the fact that the electro-magnetic field-strength tensor obliterates the parity designations implicit in the separation of E and B fields? Pauli, on p. 78 and 79 of his monograph on relativity delves a little deeper and seems to say that one may choose one of two different forms for the electro-magnetic field-strength tensor, one of which he calls the dual tensor. But the issue remains. The parity conservation rules from classical EM evaporate leaving an indeterminacy, don't they?