I am wondering if you can switch sides, and help out the other side. I'm torn from words said by NPCs of the opposing side.

During the Jagged Crown quest, you can give the crown to the leader of the other side. After that, you're stuck with the side you gave it to.

Oh, well can't I stroll into the opposition's headquarters?

Yep. it's called reloading a save.

Not after this... Of course, you could try. But I highly doubt it'll work. Any of the opposition's camps will be semi-hostile to you (they will tell you to keep walking, and attack if you stay), so I can assume Ulfric and the General will tell you to piss off as well.