Most of the credit for this goes to Ultima, since I wouldn't have even known about these if it wasn't for them. 623C corner mixups are done by spacing the 623C in the corner in such way that you'll be able land behind them and throw out 2C. Ideally your spacing will be tight and there's will be very little room between you, the opponent, and the corner.
In CMoon, it's easy as hell. Conveniently, CMoon is the one that needs it most. In C-Moon you simply end your standard BnB with 2C 5C 623C. That's really all there is to it.
In FMoon though it can be little tricky. During the dash in part of your BnB you may need to do anything for:
Cancel it early into 5B 5C 623C or 5B/C 623C
Cancel it early into 2A 623C (This is not needed to often)
Cancel it into early 623C
Now, if you're already in the corner and you want to run it, the only I've found is to delay your second j[C] quite a bit and then walk backwards a bit 2A 623C.
So, here's a pro and con:
623C mixup is much much easier in CMoon and can be quite difficult in FMoon. Of course, FMoon has 623A.