As I understand it, there is basically hidden and public information in Magic. Now let's suppose I have multiple face-down cards on the battlefield, and my opponent gets to look at one of them. Is there anything in the rules that specifies what I can do to those face down cards and what I can't? My opponent now knows what one of my face down cards is, but does he have the "right" to know which one it is, or may I shuffle them so that he no longer has access to that information? My guess is shuffling is ok, but I have no idea where I'd look in the rules...
No, you can't do that.
From Comprehensive Rules (bold mine):
707.6. If you control multiple face-down spells or face-down permanents, you must ensure at all times that your face-down spells and permanents can be easily differentiated from each other. This includes, but is not limited to, knowing the order spells were cast, the order that face-down permanents entered the battlefield, which creature attacked last turn, and any other differences between face-down spells or permanents. Common methods for distinguishing between face-down objects include using counters or dice to mark the different objects, or clearly placing those objects in order on the table.