Suppose I control a Vesuvan Shapeshifter and a face-down morph creature, and I then choose to turn the Vesuvan Shapeshifter face-down. My opponent definitely should still be able to tell which of the two face-down creatures is the Shapeshifter (I'm not allowed to shuffle them around). But that can get a bit tricky, as I may move them around in the course of the game, perhaps rearranging for attacking and blocking or anything else.
It seems that there is a mix of private and public information here: which permanent is which is public, but the identity of a face-down creature, even if it was previously revealed, is private. Does my opponent simply have to keep track and never forget which is which? Do I have to answer questions like "which one of those is the Shapeshifter?", "which one of those is the one you turned face down a minute ago?", or any other related questions, or can I refuse to answer or even lie in response to those questions?
(Note: I'm asking out of curiosity about tournament rules, i.e. what a rules lawyering player could get away with if their opponent forgets to mark the face-down card. Personally I'd happily answer the questions, same way I'd play with a revealed hand to save my opponent from having to write it down, but I know that's not required.)