Keeping track of dynamic positions of items in a grid using List/Array?

I have a 4x4 grid containing 2d sprites. When certain condition is met, a complete row or column entirely moves up/down/left/right. When this happens the last element from that row / column moves to the first position in that row / column and so on. So, when I move elements of row / column, I want a single set of objects that I can move considering all the position changes that might have happened earlier. I'm able to do it manually, but how to go about it dynamically? As you don't know which row / column will be moved by player and in which sequence.

I have following code, here Tile is a class added on a 2D sprite.

 public List<Tile> tileList;
 void Start ()
 {
 for (int r = 0; r < gridLength; r++) 
 {            
      for (int c = 0; c < gridLength; c++) 
      {
                 tempTile = Instantiate (Resources.Load ("Tile", typeof (Tile))) as Tile;
                 tempTile.transform.position = new Vector2 (c * xOffset - xStart, r * -yOffset + yStart);
 
                 tileList.Add (tempTile);
             }
         }
 }

To move tiles, I'm doing something like this -

         tileList[0].MoveRight ();
         tileList[1].MoveRight ();
         tileList[2].MoveRight ();
         tileList[3].MoveRight ();

and then moving last element ie.e tileList[3] at the first's position. Now, to move 2nd row down, I'll need to something like this -

 tileList[1].MoveDown ();
 tileList[5].MoveDown ();
 tileList[9].MoveDown ();
 tileList[13].MoveDown ();

How to keep track of these updated tiles i.e which tile is at which position and then move the appropriate tiles?

Any help would be appreciated. Thanks.