the period is 1000msec., or 1 second. When I apply a 66 hz. squarewave to RC5, I the LCD says 66
Returns the same here.
When I change the period to 2000, it says 33. Shouldn't it say 132?
Yep. Just tested one with with a 66hz signal, and it does indeed return 132.

Also, if I change the period to 500, the LCD says 33.
That's what it should return.

Are you sure you're not changing anything else when you're testing it with 2000 as the
period?