I recently married, and I have been told 2 different sides - one says the engagement ring should be on first, because I received it first, and the other says the wedding band goes on first, because this is the way most people wear them. Personally, I feel my engagement ring should be on first, since I received it first. Does it make a difference which way they are worn?
Tradition says that the wedding ring is closest to the heart. So, it would be put on the ring finger first. Supposedly there is a vein in the third finger that goes to the heart.
It is your ring though. If you want to wear it the other way around, it is your choice. Rebecca Black, Etiquette Now