'in the case where the pieces are obviously better on one of the boards,
a
mating race.'
Of course, the Kings will take shelter on the side where the pieces are
weaker.
Suppose that the left is Tripunch and the right is Shatranj; with Kings on
the right, you can use the weak Shatranj pieces to shelter them from the
grim Reapers and grimmer Combine. Kings on the left may tie down valuable
pieces in defensive roles.