Maxwell–Faraday equation symmetry violation

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I'm feeling devious.

How would you feel about the electric field if you had moving magnetic charges? Now it's the electric field that is the pseudovector.

Finally, you have a wave in free space. E and H both relate to one another via identical looking curl equations. So now which field is real and which is the pseudovector?

The answer is that just as there is space-time where space and time are mixed for different inertial frames, the same is true for E and H. As has been mentioned already, neither field alone is the complete picture. Both together (EM field tensor) is an object you can hang your hat on.