Sorting pennies in the dark

I am recurrently amazed at the wonderful logic and math puzzles posted at Futility Closet.   Here's a recent one:

You’re in a pitch-dark room. On a table before you are 12 pennies. You know that 5 are heads up and 7 are tails up, but you don’t know which are which. By moving and flipping the coins you must produce two piles with an equal number of heads in each pile. How can you do this without seeing the coins?

I was not able to solve this on my own and had to peek at the answer.  Even after seeing the answer, it took me a long time to comprehend why it works.

I'll add some thoughts in the Comments section, but for the answer, go to Futility Closet.