(This is frustrating, I've tried to correct the spelling mistakes in the above post, but it won't have it)
Is Symmetry White's Friend ?
True, White might have less winning chances than in the fianchetto variation, BUT so does Black......
How is this a good thing for white?
If white is giving up some "winning chances," then black's position has only improved. This isn't "white's friend." The only case where this would be true would be if black is rated far higher than white to begin with, but in that case, giving up the advantage in this way is tantamount to suicide.

It's better to draw 9 games and win 1 than win 5 and lose 5....(although possibly more boring :-) )
As White if I have a "solid" position that can be frustrating to play against and could possibly frustrate a younger or impatient player into making a mistake
..and I think the extent you are willing to concede chances to Black for play depend on the person's individual makeup. Fischer would agree with you perhaps but Petrosian wouldn't...

Interesting idea, but it MUST be at least one symmetry break point, otherwise the game remains symmetrical - in other words - a draw!

this is what happened to me (I am white)
1.e4 e5 2.Nf3 Nf6 3.Nxe5 Nxe4 4.Qe2 Qe7 5.Qxe4 Qxe5 6.Qxe5+ 1-0 LOLOLOL

I would say that in general quiet symmetrical positions tend to be close to equality whereas in sharp positions having the move may well bring sizeable advantage because there's a chance to attack and force the opponent into defence. Of course such generalities can be bit dangerous...

interesting idea, but how much can we move symmetrically without losing pieces and strategical advantage? Is there a way to quantify this thing?

I think breaking symmetry is best as soon as possible because you create a different set of imbalances for each side.

As some Russian GM with a sense of humor once pointed out, eventually Black has to stop copying White's opening moves because Black is not able to also copy the move "checkmate"
Obviously the point is Black has to stop being a copycat sooner or later, so it's good to know at what point (regardless of the opening played) Black should "quit while he/she is ahead" although maybe we should say "quit while he/she is behind?"
The possibilities for lowbrow humor appear endless here...

rooperi left out the last (16th) moves:
Rotlevi - Eljaschoff, St. Petersburg 1909
1.e4 e5 2.Nf3 Nf6 3.Nc3 Nc6 4.Bb5 Bb4 5.O-O O-O 6.d3 d6 7.Bxc6 Bxc3 8.Bxb7 Bxb2 9.Bxa8 Bxa1 10.Bg5 Bg4 11.Qxa1 Qxa8 12.Bxf6 Bxf3 13.Bxg7 Bxg2 14.Bxf8 Bxf1 15.Qxf1 Qxf8 16.Qg2 Qg7 (see diagram) ½-½
SOURCE: http://timkr.home.xs4all.nl/records/records.htm
PS: Tim Krabbe is a great source for chess records!
Right at the start of the game the position is symmetrical. A truism. If Black copies White, can he get any sort of equality in any/most/all of these types of positions ?
Let me start the ball rolling with one kind of symmetrical situation:
I have played the Tarrasch as Black and as White would play this OTB possibly... This is a difficult postion for both sides from the point of view of knowing what to do...but I prefer White...Black can try cd followed by Ne4 but I'm not convinced. True, White might have less winning chances than in the fianchetto variation, BUT so does Black......
More generally with symmetrical postions there will come a time when White will check Black's king and Black can't copy that !!
So perhaps some of you can posrt other symmetrical synarios with your thoughts ?
Is it the case therefore that Black's best strategy in the opening is to break the symmetry as soon as possible OR are there positions of symmetry going into the game of 10 , 15 moves etc where White is in "Zugzwang Light" ?