To say that a card is face-down, is to say the back part of the card is facing up. When a Monster Card is face-down, it can only be horizontal (i.e., in Defense Position). You can only have a face-down monster in Attack Position by the effect of Darkness Approaches.
When a Spell or Trap Card is face-down, it is always vertical.
When any card is face-down, unless specifically mentioned by a card effect, its controller is not obligated to disclose any details of that card to his/her opponent, even if that card was face-up on the field a short time ago.