I just want to be clear, I think we don't mean the same thing by Facing. Facing would be like if we had minis and someone ended their turn saying "my character is facing the direction I have my mini facing". In fact, I know that is now how the game works by default because facing is an option rule in the dmg. See below:
[ +- ] Optional Rule from DMG: Facing
Optional Rule: Facing
If you want the precision of knowing which way a creature is facing, consider using this optional rule.
Whenever a creature ends its move, it can change its facing. Each creature has a front arc (the direction it faces), left and right side arcs, and a rear arc. A creature can also change its facing as a reaction when any other creature moves.
A creature can normally target only creatures in its front or side arcs. It can’t see into its rear arc. This means an attacker in the creature’s rear arc makes attack rolls against it with advantage.
Shields apply their bonus to AC only against attacks from the front arc or the same side arc as the shield. For example, a fighter with a shield on the left arm can use it only against attacks from the front and left arcs.
Feel free to determine that not all creatures have every type of arc. For example, an amorphous ochre jelly could treat all of its arcs as front ones, while a hydra might have three front arcs and one rear one.
On squares, you pick one side of a creature’s space as the direction it is facing. Draw a diagonal line outward from each corner of this side to determine the squares in its front arc. The opposite side of the space determines its rear arc in the same way. The remaining spaces to either side of the creature form its side arcs.
On hexes, determining the front, rear, and side arcs requires more judgment. Pick one side of the creature’s space and create a wedge shape expanding out from there for the front arc, and another on the opposite side of the creature for the rear arc. The remaining spaces to either side of the creature are its side arcs.
A square or hex might be in more than one arc, depending on how you draw the lines from a creature’s space. If more than half of a square or hex lies in one arc, it is in that arc. If it is split exactly down the middle, use this rule: if half of it lies in the front arc, it’s in that arc. If half of it is in a side arc and the rear arc, it’s in the side arc.
Since this is an optional rule, this means without this rule in play players are generally looking in all directions for danger. This is the medusa can affect you if you don't look down. You can close your eyes or look down, but since the default rules don't let you say "I am only looking north", it is assumed you might look in her direction in passing if you aren't looking down or closing your eyes.
I hope this ruling isn't going to cause too much friction. This is going to be important against hythonia, as she is a buffed up medusa. I'll put it to a vote, but I guess this should have been talked about from the start. I just assumed everyone thought this way.
How should we handle a medusa's gaze?
Default rules, I have to look down or close my eyes if I don't want to make a save
Optional facing rules (see above)
Hybrid: I can say "I am not looking in the medusa's direction", but anyone fighting from the medusa's side has advantage against me