How to do random shuffling of Movie Clips by maintaining Row-Column symmetry ?

Your use of the word symmetry is likely confusing the explanation of what you are after.  If your goal is to maintain the grid formation while the letter positions change within that formation then you can do that using an array (containing the instance names) that you shuffle into a random order and some looping thru it for the placement.

Here are the two portions of coding that you will need to work into whatever you currently have...

Below is a function for shuffling an array using AS3.

function shuffle(a:Array) {
    var p:int;
    var t:*;
    var ivar:int;
    for (ivar = a.length-1; ivar>=0; ivar--) {
        p=Math.floor((ivar+1)*Math.random());
        t = a[ivar];
        a[ivar] = a

;
        a

= t;
    }
}

// Usage:
shuffle(letterArray);

For the placement you can do it using one or two loops.  Here is a two-loop approach....

var count:int = 0;

var rowY:Number = 0; // replace with starting y value

for(var i:int=0; i<5; i++){   // for each row

      var colX:Number = 0; // replace with starting x value

      for(var j:int=0; j<6; j++){  // for each column in the current row

             letterArray[count].x = colX;

             letterArray[count].y = rowY;

             colX += distanceBetweenXs;

             count += 1;  // increment for the next letter in the array

       }

      rowY += distanceBetweenYs;

}