The front side of a double stitch is recognized by the bump at the top of the double stitch, including the base of each picot. The back side of tatting is recognized by the lack of the bump.
How can one make all the bumps show up on the front side of your work? An example: 2 ds p 2 ds cl rw. Since you have reversed your work you simply reverse the order of the double stitches and the bumps will come out on the front of your work.
Have you noticed that with needle tatting the tatting stitches are actually the same as knit and purl in knitting?