Didn't the U.S.A and the Confederates go to war with each other; how can you support the U.S and the Confeds?

This is a serious question: I see and hear people supporting the Confederacy all the time and I find nothing wrong with supporting your own government. My issue is: How can you consider yourself a confederate and still support the U.S.? Weren't their ideas so different that they went to war with each other at one time in history? To me at least, each government supports completely opposing ideas and I don't understand how you can consider yourself both. My intentions are not to offend anyone, I know there are still a lot of Confederates living in the U.S. and I am only asking this because I truly don't understand. Anyone have a legit answer?