Remember that the Superku videos on youtube are videos of my own sidescroller not of our team project.

I have an idea:

The player is a divinity that is worshiped in one period of time but persecuted in the other age for something that he seems to have done. This explains why the level is deformed and destroyed in the second time period. He can talk, trade and buy things from people but their descendants haunt him in his "dark ages", so he has to find a way to undo this crisis.
When he travels through time, at first via portals, some levels later with an amulet (or some other item/ hourglass/ ...), it will take some time to recharge his device (f.i. 2 minutes, an hourglass seems to fit nicely when I think about it) and in this time period the player has to flee from his persecutors.
If you have ever played Metroid Fusion, you know how exciting this can be (escape from SA-X, "Someone should remake the game to be playable for xbox360/ps3/wii . Cause this game is the true definion of a horror game.").

I would say we limit the gravity change ability to 180° degree turns. Now we can add 90° curves in some corners that the player can walk on, the gravity adapts to his orientation. Hard to explain, have a look at my protoype video, there is a curve at the top right at the beginning.

EDIT: The bad people could invent a new god/ creature that is supposed to protect them and to hunt the player when the game progesses (only in the future).