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Spoiler:

The key to this problem is set theory. Cards are of two types (face-up or face-down) and the real trick is understanding that if you take ten items out of the whole set the number of special cards is the inverse.

If I pick up ten cards and three are face-up then seven are face-up in the rest of the pile, which means if I take those ten cards I pick up I turn the whole pile over and both piles now have seven face-up cards. The trick is to take ten of the whole set and invert them. It always works no matter how many face-up cards I hold in the set of ten.

0 - 10
1 - 9
2 - 8
3 - 7
4 - 6
5 - 5
...

It works because there were ten and I select ten. As long as I keep my picked pile that I turn over the same size as the number of face-up cards, I can get the same number of face-up cards.

I originally thought the piles needed to be the same size, but that isn't specified in the question.