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My reflection on these games has led to the conclusion that Symposium
Chess, named for a story told in Plato's Symposium, is the variant that
most closely approximates Chess while having perfect bilateral symmetry.
Almost any Game of Chess can be played as a game of Symposium Chess. So
can almost any game of Sinister Queens Chess. A game of Symposium Chess
could also play out as the mathematically equivalent mirror image of
either Chess or Sinister Queens Chess. White's first move cannot
determine which of these the game will be played as. After White's first
move, all four possibilities remain. So each move White can make gives him
about four times as many possible games as the first move in Chess makes
possible. But half of these are mathematically equivalent to the other
half, and it turns out that half of White's moves lead to games
mathematically equivalant to what would follow from the other half. If we
consider only the moves available to White on one side of the board, say
the left side, each possible combination of King and Queen remains
distinct, so that each move leads to four times the possible games as a
move in Chess does. And if we consider the moves available on the other
side, the same will be true, but each game in one set will be paired with
a mathematically equivalent game in the other set. So White's first move
let's him choose from only about twice as many possible games as he has
to choose from in Chess. If White had 40 opening moves to choose from, he
would have more control over the course of the game on his opening move,
but with only ten mathematically distinct opening moves available,
White's first move exerts less control over the course of the game than a
first move in Chess does.

Besides taking away some of the control White has on his opening move,
Symposium Chess gives more control to Black. Both players have the power
to differentiate their Monarchs into King and Queen. This gives players
power not only over the course that the game will take from there but also
power over the significance of previous moves.

Now, although most games of Chess can be played as games of Symposium
Chess, an actual game of Symposium Chess would probably play out
differently than a game of Chess, because it does change the dynamics of
the opening game. Until a player differentiates his King and Queen, his
opponent doesn't have a clear target for his attack, and if he does pick
one Monarch as the focus of his attack, that just increases the odds that
this Monarch will be the Queen and not the King.

Overall, it would seem that Symposium Chess puts White and Black on a more
equal footing, so that White gains less of an advantage from moving first.
But questions remain to be asked. How great a difference is it? What
significance does it make? Will the difference be seen only by
statistically comparing game scores of top players? Does this difference
make Symposium Chess the better game? If so, how much better?

My inclination is that it makes only a slight difference, and that if it
is the better game, it is not better by that much. I would not presume
that Chess is junk for being asymmetical while Symposium Chess, which is
not that different from Chess, is so much greater for being symmetrical.