It is a big deal to the Japanese. The right goes over the left on DEAD PEOPLE! Unless you are training at a dojo in which the "living dead" are practicing, then I would not suggest that you get this one wrong.
It's what I was told by my sensei in Japan, and my wife, who is Japanese.
Interestingly, I'm sure I saw a Seagal movie one time in which he has his "kimono" silk top done up the wrong way (he was in a boat going up a river in Thailand I think).....maybe I was just seeing things......