The axis of symmetry may be parallel to a row, or parallel to a file (and we usually call it vertical axis), or parallel to a diagonal line. There is also the case of symmetry around a point.
From the Dawson's analysis we gather a few observations :
1. In an apparently symmetrical position, it is possible for the White to have some abilities on the one side (let us say : a move to a side file, a castling move, etc.) which are not available on the other side of the position. This means that the solution is probably non-symmetrical.
2. In an apparently symmetrical position, it is possible for the Black to have different abilities on each of the two sides of the position, resulting in a non-symmetrical solution.
3. In a diagonal symmetry, the Pawns do not have equal abilities of movement on the file or on the row, thus the solution may be non-symmetrical.
When we study a symmetrical position, we decide whether we will maintain the symmetry with the key, or we will break the symmetry of the position.
Let us see the Problem-162, by (the famous in Greece troubadour of rebetico songs) Mr. Nikos Pergialis.
 | (Problem 162) Nikos Pergialis, Newspaper "Eleftherotypia", 17/12/2006 Mate in 2 moves. #2 (7+9) |
| [8/3K4/2pBp3/p2k2p1/1ppPpp2/3R4/S5S1/3B4] | |
Tries: {1.Bc5? e5!}, {1.Be5? c5!}, {1.Sxf4+? gxf4!}, {1.Se3+? fxe3!}, {1.Sc3+? bxc3!}, {1.Sxb4+? axb4!}.
Key : 1.Rd2! (zz). If Black breaks the symmetry, he is lost.
1...a4 / b3 / c3 / c5 / e5 / e3 / f3 / g4
2.Sxb4# / Sc3# / Bb3# / dxc5# / dxe5# / Bf3# / Se3# / Sxf4#
This setting may be moved one file to the right, and the solution will be similar. (When we decide for the final form of our composition, we take care to have more pieces on white squares (or else we transpose the position) in order to be nicely-looking when it will be printed. Here the composer put on white squares 10 from the 16 pieces).
But this setting must not be moved one row upwards!
Do you see the changed detail which leaves the problem without solution?
It is the pawns above the upper Bishop, which, when they are standing on row-7, they can move with double step and thus they can interfere with the checking move of the lower Bishop (so, there is no mate in 2).
In the next Problem-163, by Dawson, it is obvious that if Knight leaves e3, the black King can move, either to d5 with flight to e4, or to f5 with flight to e4, so with the Knight on e8 at the right moment White can mate.
 | (Problem 163) T. R. Dawson, Newspaper "The Times", 23/12/1920 Mate in 6 moves. #6 (9+6) |
| [8/8/1p2p2p/1P2P2P/2pPkPp1/2P1S1P1/4K3/8] | |
Tries : {1.f5? exf5!}, {1.d5? exd5!}, {1.Sxg4? Kf5!}, {1.Sxc4? Kd5!}, {1.Kf2? Kd3!}, {1.Kd2? Kf3!}.
Key : 1.Sc2! (zz, zugzwang).
And how is the Knight going to e8?
By the left side, stepping on the edge file, outside symmetry.
1...Kd5 / Kf5 2.Sb4(+) (zz)
2...Ke4 3.Sa6 (zz)
3...Kd5 / Kf5 4.Sc7 (zz)
4...Ke4 5.Se8 (zz)
5...Kd5 / Kf5 6.Sf6# / Sd6#
If we move the setting one file to left, the Knight must go to e8 by the right side.
If we move the setting one row upwards, the problem is ruined, because the white pawns can achieve solution in four moves (because e is promoted with check).
(The position has only 7 from 15 pieces on white squares, but the composer has preferred to leave the Kings on their initial file-e).
In the miniature Problem-164, by Stavros Iatridis, we see that the key maintains the symmetry, and the two variations end with echo mates. (Please compare with Problem-45, which has diagonal symmetry and similar echo mates).
 | (Problem 164) Stavros Iatridis, Mate in 2 moves. #2 (4+2) |
| [8/8/4Q3/4p3/3SkS2/8/4K3/8] | |
Tries : {1.Qd5+? / Qc6+? / Sb3? / Sc2? / Sf3? / Sf5? / Sc6? / Sb5? / Kd2? Kxf4!}, {1.Qf5+? / Qg6+? / Sd3? / Sg2? / Sh3? / Sh5? / Sg6? / Sd5? / Kf2? Kxd4!}, {1.Qxe5+? Kxe5!}.
Key : 1.Qe7! (zz).
1...Kxf4 / Kxd4 2.Qh4# / Qb4# (echo-mates)
The Problem-165, by Carpenter, has similar position with the previous one but its solution is non-symmetrical.
 | (Problem 165) George E. Carpenter, Dubuque Chess Journal, 1873 Mate in 2 moves. #2 (5+2) |
| [4B3/8/4Q3/3SpS2/4k3/8/8/4K3] | |
Tries : {1.Bb5? / Qc6? / Kd2? Kf3!}, {1.Bh5? / Qg6? / Kf2? Kd3!}, {1.Qxe5+? Kxe5!}.
Key : 1.Qa6! (zz). The Queen needs to reach squares c6 / g6 / e2 and steps on the side file removing one flight but abandoning the guarding of the two Knights.
1...Kxd5 / Kxf5 / Kf3
2.Qc6# / Qg6# / Qe2#
The moves of the wQ to the squares c6 / g6 are seen as tries and also when it gives mates from there.
Many problems have been composed with symmetrical position. Some resemble trees, some are sketches of musical organs, some have the shape of a letter.
Some of these problems are very difficult puzzles. (Remember that the Problem-155 needed retroanalysis in order reveal which of the two en-passant captures was the key).
We will see now the Problem-166, by Anderson, which has diagonal symmetry. The Pawn c6 can move to c5, but it cannot move to d6.
 | (Problem 166) William Anderson, Honourable Mention, Ideal Mate Review, 1984 Helpmate in 5 moves. h#5 (2+4) |
| [R7/8/2p5/3k4/4q3/8/6r1/7K] | |
Key : 1.Rg1+
1...Kh2 2.Rc1 Kg3 3.Qe2 Kf4 4.Rc5 Kf5 5.Qc4 Rd8#
We close this presentation with a nice original problem, by the known chess player (ELO 2225) Mr. Emmanuel Pantavos. Here the symmetry is around a point (a unique point which is the middle of the distance of any pair of pieces similar in value but dissimilar in color).
 | (Problem 167) Emmanuel Pantavos, original, 05/05/2008 Helpmate in 2 moves, Duplex. (2 solutions) h#2 duplex 2111 (4+4) |
| [8/5p2/8/1S2kr2/1RK2s2/8/1P6/8] | |
Black plays...
1.Sd5 Kd3 2.f6 Re4#
1.Ke4 Rb3 2.Re5 Sd6#
White plays...
1.Sd4 Kd6 2.b3 Rc5#
1.Kc5 Rf6 2.Rc4 Sd3#
(This post in Greek language).
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