Hi to All,
I stand on the left-side of the wheel and it's spinning counter-clockwise. I move to the right-side of the wheel and it's now spinning clockwise. I stand on the left-side of the ballroom and the dancers are moving counter-clockwise. I then stand on the right-side of the ballroom and they are still moving counter-clockwise. Will some one please tell me; what's going on!?

The reason is that in one case(the wheel) you're moving through the plane in which the wheel is moving, whereas with the dancing, you stay in the same plane.(i.e. above the floor)
Now, if the dance floor were of glass, you you got below it and looked up at the dancers, they would be rotating round the floor clockwise.
Here's an interesting point: If the axis of the wheel is horizontal (as i assume it is)and, let's say aligned north-to-south, then, if you're to the north of the wheel, you look to the south to see it rotating. When you go the other side of the wheel, you look to the north to see it rotating, and this is the fundamental reason for the apparent change of rotation. If you went to the south, and, instead of looking north, you looked to the south into a mirror (so you could see the wheel) the direction of rotation would be unchanged.
If you have a car sticker which reads correctly from outside the car, and you look at it from the inside, it reads backwards. But look at it using the driving mirror it reads ok again. You're viewing it by looking in the same direction as you did when outside the car, even though you've moved through the plane of the sticker.
How does this help our dancing? Not at all unless you can't do a reverse turn, and try to give the judges mirrors to watch you doing natural turns (you'll have to change hold, too, of course)

Thanks Phil.
I'll have to read that a few times for it to sink in. The good thing about it is I maybe able to get to sleep, now . Happy New Year, Phil.