The question
Which way up the moon hangs
Why is the same moon mirrored when you look at it from Australia?
Two people on opposite sides of the Earth see exactly the same fraction of the Moon lit at exactly the same second. They do not see the same picture.
The same moon, two pictures
Take one instant — say the evening of 15 June 1990 — and look at the waning gibbous moon from Amsterdam. It is lit down its left-hand side. Now look at the same moon in the same second from Sydney. Same phase, same percentage, same everything the numbers can express. It is lit down its right.
Neither observer is wrong and nothing about the Moon changed in between. What changed is the two of them.
You are standing on the other side of a ball
The direction you call up is the direction away from the centre of the Earth, which is a different direction depending on where you are standing. Someone in Sydney has their head pointing roughly opposite to someone in Amsterdam. They are, to each other, upside down.
So when both of them turn to face the Moon, they are holding the same object against opposite ideas of vertical. The Moon has not rotated. The frame it is being described in has flipped, and a flipped frame turns a left-hand crescent into a right-hand one.
This is the same reason the sky chart on this site puts east on the left. You are looking up at a dome from the inside, not down at a map from above, and half of what feels wrong about astronomical drawings is that inversion.
Near the equator it lies on its back
Between the two extremes there is a continuous turn rather than a switch, and in the tropics the crescent can end up lying almost horizontally — a bowl, or a smile, with both horns pointing upward. It has a folk name in several places: the wet moon, or the Cheshire moon.
The reason is that the lit side always points at the Sun, and the angle at which the Sun's path meets your horizon depends on your latitude and on the time of year. Near the equator that path rises steeply, close to vertical, so a crescent just after sunset has the Sun almost directly below it and is lit from underneath.
What most drawings get wrong
A great many moon-phase illustrations, including a good number sold as personalised prints, draw one orientation and give it to everybody. That is correct for roughly the half of the planet the illustrator lives on and mirrored for the rest.
It is an easy thing to miss, because a moon phase feels like a single global fact. The fraction lit genuinely is one: everyone alive sees the same percentage at the same moment, which is why this site's readings do not ask where you were. Only the drawing does.
What this drawing still does not do
The instrument here turns the disc by hemisphere, which captures the large effect — the mirror flip that makes a print wrong for half its audience. It does not model the continuous tilt.
That tilt is real, and it moves during a single night: the Moon rises with its lit side at one angle and sets with it at another, because the whole sky is rotating around you as it goes. Drawing it would mean choosing an hour and asserting it, and a poster is not a clock. So the disc here is drawn as it appears when the Moon is high — the orientation you would see if you went out and looked at the best moment of the night — and this paragraph exists because a limit you do not mention is a limit you are hiding.