White KIA meets black KID

There is an old Petrosian game where he won as black after keeping the symmetry for 8 moves (1.Nf3 Nf6 2.g3 g6 3.Bg2 Bg7 4.d3 d6 5.e4 e5 6.Nbd2 a5 7.a4 Na6 8.Nc4 Nc5). White soon started to play on the kingside and black played in the centre, and the central strategy prevailed. This raises the question of symmetry - isn't it so that the initial postion is also symmetrical, and that any symmetrical position can be more or less advantageous for the player to move. In this kind of closed position it could easily be that it is almost easier to play black, since you can react appropriately depending on the move white choses. I believe that in the position noted above the value of the extra tempo is smaller than in many others, thus the position is probably already balanced.