You can't go to both of these interviews - it's impossible to be in two places at once. You will have to reschedule one of them.
You do not need to mention that you are rescheduling because of another interview - just pick a company and tell them you have an appointment you can't miss, and ask them to reschedule. If they refuse, ask the other company if they can reschedule.
If neither company will reschedule, call and cancel the interview you will miss the least - or just don't show up for it, if you cannot handle cancelling the interview. But if neither one of them will reschedule, it is the polite thing to do to cancel at least one so that you leave them on good terms.
Call as soon as possible - it improves the company's ability to reschedule for you, and is much more courteous and convenient to their needs as well.
As for saving face, the effect of asking to re-schedule an interview is very minimal in terms of losing respect from your employer, both for the interviewer and for the HR department. You should be courteous and polite in requesting to re-schedule, understanding if they cannot accommodate your schedule, and apologize for the inconvenience at most.
This really isn't a huge issue - scheduling mishaps happen all the time. It's better to own up to it now and ask to reschedule than it is to stew over the mistake and make things more difficult for HR and your potential boss by not correcting the mistake.