What is it called when the reverse of a coin is on the obverse as well?

I have a 1887 Morgan Silver Dollar. It is in very good condition with nice luster. I noticed from my other silver dollars it looked different. (I am not a coin expert, I just learned a little terminology and examined it closely under a microscope.) When I looked under the microscope I saw different shades and rises in metal and lines. When I finally realized that I could see a faint outline of the eagle from the back I tried to research it and figure it out. But I have not found any info on it. Could someone help me with this, and could it be valuable? The coin has a double die on the date, also on the lips there is a very strong die of a twig that wraps around the eagle on the back, you can see the middle line and the shape excellent. Also there are letters that are faintly marked on it that looks like a mirror image.