It is said that painting the left eye of a Daruma doll first is the correct way.
There are a few reasons to support this idea. When a person has no equal, it is described in Japanese that there's no one who stands on that person's right. Also, left-side seats are considered higher in rank in a seating arrangement. However, there is no correct order of painting eyes of a Daruma doll.
Drawing eyes signifies "opening of a Daruma doll's mind's eye", not just filling in the blank eyes. To paint Takasaki Daruma's eyes is to put its soul into it.
In the olden days, round, wide-open eyes were painted by the manufacturer and sold. In modern times, buyers of Daruma dolls do their own painting.