I think you are on the right track putting an emphasis on symmetry to solve
problems involving fairness (and balance) for chess variants.  Obviously,
there are several factors that determine the complex functioning of a game
which affect fairness- more than just geometric symmetry of the pieces (and
board) ... as Thompson correctly points-out.  From my own positive
experiences in refining Hex Chess SS, I am confident that if your efforts
are thoroughgoing and if you are willing to break with tradition to a
sufficient extent with your game designs, you will achieve a level of
fairness that satisfies your high standards.

Some problems impacting fairness are traceable to using the traditional
white-black turn-order in chess variants.  The first-move-of-the-game
advantage (for white) is too high in relatively-small games such as
standard Chess.  Of course, this problem can be reduced by the use of more
pieces per player, more limited-range pieces instead of unlimited-range
pieces, a larger board, etc ... but never solved and eliminated as a game
becomes ridiculously large overall and unplayable for people.

The white-black-black-white turn-order has passed every test I have thrown
at it.  The predestined unfairness generally characteristic to turn-based
chess variants [With theoretically 'perfect play' by both players, one
player is certain to win despite the unsurpassable gameplan of the other
player in Hex Chess SS.] is so very small, I am unable to pinpoint whether
it actually favors white or black (although I hold the opinion that it
probably favors white).  So, I have abolished draws as an admissible
game-ending condition while maintaining a higher level of fairness than
exists for the vast majority of chess variants.

In case you are wondering, conclusive endgames are always achievable in
chess variants having the appropriate pieces starting the game and the
appropriate board geometry.  This so-called 'pendulum turn-order' is
symmetrical and fair to black in ways that the white-black turn-order
provably fails to be.

These two statements have complicated explanations available here:

Description
Symmetrical Chess Collection
http://www.symmetryperfect.com/shots/descript.pdf

I recommend reading the entire 52-page essay to better understand the
following sections.

See section 14 (pages 20-23).
'minimizing the first-move-of-the-game advantage'

See section 20 (pages 36-41).
'game-ending conditions'

Living proof exists in all 3 variants of the working game Hex Chess SS
which requires the Zillions Of Games program to play:

Symmetrical Chess Collection
http://www.symmetryperfect.com/shots