99.9999% of the population never notices this, of course; but roughly half the planet (Mayflies) believes that there’s a May 31st, and the other half of it (Junebugs) believes that there’s a June 31st. Why? Because of a fairly impressive magic spell, of course. More on that later.
The effects are remarkably comprehensive, and mostly involve altering people’s perceptions so that it never registers that that person or document is currently referencing a May/June 31st. If a Mayfly encounters a Junebug using the ‘wrong’ date, or vice versa, it doesn’t seem very important, and it won’t get saved to long-term memory. The same thing happens when it’s an incorrect physical reference, too. The brain assumes that there was a screw-up somewhere, and then it forgets about it. And nothing about the phenomenon ever seems to get written down, either. The only people who do know about this situation are typically mages of no small power, and they generally benefit from the May/June 31st thing, so they’re not going to rock the boat.
What’s the advantage of this, then? Well, magic runs on contradictions and ambiguity. In this case, creating an entire month where two groups of people have contradictory opinions on which day it is and both groups are right generates a lot of power. Magical workings are simply more powerful on the 31 days between May 31st and June 31st (inclusive) — or is it really 32? The math only works when it’s subjected to a very strong magical field. Which is to say, one month out of the year.
It’s all really benign, honestly. Most people don’t even notice any oddities. And there is an entire organization that operates during the month-plus of June to make sure that people don’t notice anything derived from those oddities, like fireballs in the street or zombie assassins in the boardroom. Because, of course, not every person given great power is willing to accept a commensurate amount of responsibility to go along with it.