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I have a simple grid puzzle:
Example:

| X | Y | X | 
| Y | X | Y | 
| Y | X | Y | 

The move is defined as a row or a column move. So when you move one cell, whole column/row moves.
Example top row right 2 steps:

1 step
| X | X | Y | 
| Y | X | Y | 
| Y | X | Y | 

2 step
| Y | X | X | 
| Y | X | Y | 
| Y | X | Y |

Now column move will look like this on the last row move up

1 step
| Y | X | Y | 
| Y | X | Y | 
| Y | X | X |

and so MoveRight - the most right cell will become the most left cell etc...

I am wondering if it is possible to go to every possible grid combination only with this moves. Let's assume that the grid will be 3x3 but it can be any size.

So, for example, I create random grid 3x3 with the same X count and Y count like this:

| Y | Y | Y | 
| Y | Y | X | 
| X | X | X | 

Is it possible to check if there is a solution on how to make this grid the first grid with already specified moves? Or how to approach this problem?

Thanks, everyone.

In the general case (all symbols allowed), you cannot reach every position. The argument is a parity argument, similar to how half of the configurations of a Rubik's Cube cannot be achieved.

Every legal move in your game is equivalent to an even number of "swaps". Say you start with

ABC
DEF
GHI

and move the top row right:

CAB
DEF
GHI

This is equivalent to swapping BC, and then swapping the resulting AC:

ACB
DEF
GHI

CAB
DEF
GHI

Every move you can make is similarly equivalent to two swaps. This means that every position you can possibly ever arrive at is the equivalent of an even number of swaps.

The result of this is that you cannot ever arrive at a configuration that requires an odd number of swaps, like this one:

BAC
DEF
GHI