The card Darkness Approaches has the following text:
Discard 2 cards from your hand. Select 1 face-up monster and flip it face-down, but do not change its battle position.
Because it specifically says to not change its battle position, this would allow a face-up Attack Position monster to become a face-down Attack Position monster.
How does this work exactly? If a monster attacks a face-up Attack Position monster, what happens? Does it flip in response to the attack? Can I Flip Summon a face-up Attack Position monster? Are there other oddities that I need to be aware of?
Sometime in mid-2017 (after this question was originally asked), the card text for Darkness Approaches was errata'd to no longer allow face-down Attack Position monsters:
Discard 2 cards from your hand. Select 1 face-up monster and change it to face-down Defense Position.
Still, it's interesting to keep this question around for historical purposes.