Because you would get the same solution as before, just with the E and B fields switched.
Speaking of whether magnetic monopoles exist:
I always thought magnetic monopoles were an interesting topic for discussing the scientific method with students. For example, how many magnetic monopoles would you have to see to consider that magnetic monopoles exist?
Most people answer 1. But historically, searches for magnetic monopoles have seen signals (there are two in literature I believe). But the signals were not repeatable, so we don't consider magnetic monopoles to have been discovered.
There are some theoretical niceties that we get if magnetic monopoles exist.
Wouldn't it be frustrating if we later find out they exist, and they are just rare enough that the original 'discovery' was just discarded? There will always be experiments which keep looking. Who knows, but I doubt they will find anything.
So how many magnetic monopoles would you have to see in the lab before you'd believe it? :)