Several people have asked me to illustrate further just what the practice of combining two coats of arms together on one shield via the antiquated method of dimidiation looks like. This form of marshaling was used extensively in the Middle Ages but fell out in favor of impaling. With dimidiation, as the name implies each of the two coats of arms is split down the middle and only one half of each is depicted in the two halves of one shield. An excellent example of this are the arms of Dolní Lomná, a village in the Czech republic. The arms depict an eagle on one half and a tree on the other. Dimidiation obscures part of each of the coats of arms and also sometimes creates somewhat unusual looking creatures like this demi-eagle/demi-tree or even more amusingly the famous arms of the Lord Warden of the Cinque Ports who seems to bear lions that are also half boat!
EXARANDORUM 
What determines which half goes where? Aesthetics or some other determining factor?
As I mentioned in my post about the coat of arms of the new archbishop of Dubuque, dimidiation fell into disuse in favor of impalement.
The same principle holds true for dimidiation. That is the “senior” half is in the dexter and the “junior” in the sinister. In the case of a married couple the senior would be that of the husband. In the case of ecclesiastical arms the senior would be that of the diocese.
I went back and read the previous post. Fascinating – thank you.