How can I reposition a soldered copper 90-degree fitting
You can indeed desolder fittings. However, as BMitch points out, you'll have to remove the fitting, clean it, and solder it in again.  

If you heat the solder up enough, the fitting should slip right off.  If it doesn't come off easily, it's not hot enough. Try to spread the heat around, so you heat the joint evenly.  

You'll want to use a pair of pliers, to remove the hot fitting. Do not touch it with your hand.