What if we elected our presidents like this (see details)?

Disclaimer: not suggesting this as a serious thing, just want to imagine and discuss.

Every 4 years we rotate between a conservative and a liberal president.

Example: Obama, a liberal, just got his turn, so the next prez has to be conservative. We hold basically a Republican primary, but everybody votes in it, and winner is prez. In four years we do the same thing with the Dems.

Advantages I thought of: everyone gets their turn, no tyranny of the majority, major checks and balances ensured, we probably end up with moderates from each party instead of crazy people.

Disadvantages I thought of: inability to respond to the needs of the times, people trying to game the system by pretending to be one party but really being the other, progress being made and then immediately undone constantly.

What do you imagine might happen? Both good and bad possibilities, please.

Again, just a thought experiment.