Even someone who has never seen a minute of an NBA game would be transfixed watching Stephen Curry play basketball—but maybe not because of his otherworldly shooting or because the league’s most valuable player looks like a teenager. This person would probably ask something more obvious about Curry: What’s he doing with his mouthguard?

Curry’s mouthguard acrobatics are normal in the NBA by now. But that doesn’t make them any less baffling. More often than not, when he steps up to the free-throw line, Curry can be seen chewing...