#C#, self-inverse, no randomness#

If the original image has dimensions that are powers of two, then each row and column is exchanged with the row and column that has the reversed bit pattern, for example for a image of width 256 then row 0xB4 is exchanged with row 0x2D. Images of other sizes are split into the rectangles with sides of powers of 2.

namespace CodeGolf
{
    class Program
    {
        static void Main(string[] args)
        {
            foreach (var arg in args)
                Scramble(arg);
        }

        static void Scramble(string fileName)
        {
            using (var origImage = new System.Drawing.Bitmap(fileName))
            using (var tmpImage = new System.Drawing.Bitmap(origImage))
            {
                {
                    int x = origImage.Width;
                    while (x > 0) {
                       int xbit = x & -x;
                        do {
                            x--;
                            var xalt = BitReverse(x, xbit);
                            for (int y = 0; y < origImage.Height; y++)
                                tmpImage.SetPixel(xalt, y, origImage.GetPixel(x, y));
                        } while ((x & (xbit - 1)) != 0);
                    }
                }
                {
                    int y = origImage.Height;
                    while (y > 0) {
                        int ybit = y & -y;
                        do {
                            y--;
                            var yalt = BitReverse(y, ybit);
                            for (int x = 0; x < origImage.Width; x++)
                                origImage.SetPixel(x, yalt, tmpImage.GetPixel(x, y));
                        } while ((y & (ybit - 1)) != 0);
                    } 
                }
                origImage.Save(System.IO.Path.GetFileNameWithoutExtension(fileName) + "-scrambled.png");
            }
        }

        static int BitReverse(int n, int bit)
        {
            if (bit < 4)
                return n;
            int r = n & ~(bit - 1);
            int tmp = 1;
            while (bit > 1) {
                bit >>= 1;
                if ((n & bit) != 0)
                    r |= tmp;
                tmp <<= 1;
            }
            return r;
        }
    }
}

First image:

Second image: