I've done some thinking about puzzleshipping these last few days, and have come to the conclusion that I think of them as soulmates, rather than just in love. (This is a change from when I was like, ten and didn't have much perception of relationship depth or complexity. I am now an adult and my Yu-Gi-Oh obsession has come back, which has led me to do some thinking.
They are two souls completely intertwined, able to live without each other but never able to get rid of the aching hole left by their other half. They compliment each other. They complete each other. They are Partners. They are their other selves.