Here's a guess, and this is only a legit guess if you can partly but not fully insert the skewer. The axle is bent, near the drop out, on the fixed side. When inserting the skewer, if it goes in maybe a centimeter but then stops, that would be my guess. It would work the other way because... I don't actually know the mechanical reason, but let's just say I've seen it a number of times. I think it's cause you can get a run at it, a longer shove before it hits the bend. Someone smarter than me can tell us the real reason.
Just run your QR backward and give people dirty looks if they call you out for it. Seems pretty easy.
-james