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We seem to have gotten a wide range of responses here, from being as
symmetric as possible to de-emphasizing symmetry and looking at other
options to balance the sides. I actually like all the answers given here,
and use or will use each in designs. 
Derek - I read your paper [again] before replying here. I find that a very high level of symmetry does allow for a fair game, though I have designed
most of my games without '4-axes' symmetry [both E-W & N-S]. Chieftain Chess, which I think is my most fair design, uses 4-axes symmetry, and
breaks rather completely with traditional chess in having no pawns or
other uni-directional pieces, multi-move turns [up to 4 pieces per side
per turn], all shortrange pieces [1, 2, or 3 squares maximum], a 'large'
[12x16] board, no king but rather 4 chieftains, and leadership rules. I'd
considered having white start the game by making only 2 moves, and thus
balancing the game, as each side, when it's finished its turn, has moved
2 more pieces than the opponent, but I believe the other design features
make this superfluous, to the extent I did not wish to add another unique
rule to the game. I believe this is a perfectly fair game for both sides
as is, because of the initial placement/separation of the sides, the short range of piece movement, and the ability of black to move backward away
from contact and into a stronger defensive position. Leadership rules and
the unique nature of the inclusive compound pieces add to the
effectiveness of the placement, range, and [backward] mobility of the
pieces. And while it may not be perfect, it can be played beginning to end with constant E-W symmetry at the beginning and end of each half-turn [each player's move]. I invite your comments on this game.