Certainly you were making a joke, but I’ll offer the explanation.
Switch hitting is relatively common compared to pitching with both arms. It is advantageous to hit from the opposite side of the plate than the pitcher throwing, because it turns out that it’s easier to hit a curveball or slider that’s curving toward you rather than away from you.
The only rule that really governs that situation is that the hitter may not make a mockery of the game by switching sides over and over. It doesn’t really comment on what the pitcher may do, though ultimately, the umpire may adjudicate any situation not covered in the rulebook as he sees fit. Thus, at some point, the umpire stopped the hitter from switching sides while the pitcher was allowed to continue.
There actually is a rule for this. The pitcher must declare which side he will pitch from prior to the batter deciding from which side he will hit. The pitcher must also use that same throwing hand for the entire at bat.
Apparently, there is not such a rule- or at least, neither ESPN nor the executive director of the Professional Baseball Umpire Corporation are aware of it. That rule does exist in the MVC, which I’m assuming is where the pitcher played college ball, but MLB is still working on their response.