I had a saved game that I loaded up, where I had a lot of influence and many planets within my area that were flipping (at least one or two per turn). I made my moves and clicked turn, and nobody flipped. Something came up and I had to quit, so I saved and came back to it later. Same thing, made my moves and clicked turn. Again, nobody flipped. I got curious, so I saved, quit, and reloaded. Did another turn, and again no flips. I then noted that none of the enemy planets in my sphere of influence show the "rebellion" skull until I click turn. Also, I have noticed that the influence borders are a little wierd (not where you expect them to be) until you have clicked turn.
So is it not possible to flip planets on the first turn after a load, or did I just have some bad luck?