Which way does the shadow move across the moon?

Shadows can cross the moon either from right to left or from left to right, depending on the source of the shadow. Shadows caused by the moon's phases pass from right to left relative to an earthbound observer. The shadow of an eclipse passes from right to left.

Over the course of a month, the area of the moon covered by its own shadow traverses from right to left over the course of a lunar month.

Twice a month, the moon's orbit carries it across the Earth's orbital plane. When this passage occurs during a full moon, the Earth's shadow crosses the disk of the moon from left to right as seen from Earth.