One thing I've noticed is that Link monsters don't even have defense points. A card I like to use is Book of Moon, which lets me switch a monster into face-down defense mode. Since Link monsters don't even Def category, is it still possible to switch them into face-down or face-up defense?
From the Official Ruling, page 15, first paragraph:
Link Monsters have no DEF and cannot ever be in Defense Position. They can’t be changed to Defense Position by a card effect. They can’t even be flipped into face-down Defense Position.
We can see that Link monsters can't ever be in Defense position, not face-up or face-down. This also excludes changes from card effects like Book of Moon.
More info on this answer and question.
Link monsters cannot be in defense mode as they do not have any defense points, also changing to defense mode changes the zones they point to as the card orientation has changed.
Please see these links for more information.
You can use a card called Darkness Approaches which allows it to be face down either attack or defense relevant to the card's current position.
More on the rulings of: Darkness Approaches: Ruling Very useful to flip the Link monsters to face down attack position. Hopefully, you can add these can of worms into your deck for maximum trolling.