I worked out what was the actual problem with your diagram: not enough connections. The double-digit Faces have six each, and all the rest have four. There should be eight, two for each side of the square.
But going on your descriptions, I have these as White's and Black's starting areas.:
If I got this right, then it does have an unacceptable asymmetry: Black's connections along the inner cube are all double digits, while White has different digits on all eight of its neighbors. I don't think any other game has that level of difference between the two sides.
I worked out what was the actual problem with your diagram: not enough connections. The double-digit Faces have six each, and all the rest have four. There should be eight, two for each side of the square.
But going on your descriptions, I have these as White's and Black's starting areas.:
If I got this right, then it does have an unacceptable asymmetry: Black's connections along the inner cube are all double digits, while White has different digits on all eight of its neighbors. I don't think any other game has that level of difference between the two sides.