Green Card and Passport names on Given and Surname

Since my surname and Given Name were flipped, my GC came the same way. I changed the passport to reflect my correct given and surname after the GC approval. Since changing the name again on GC is an expensive option I can wait for name change during naturalization. My question is if I travel outside US is it a must that the given and surname on passport match GC while coming back? Can I have the birth certificate and other credentials as a back up?