You can put any fabric you want on the back, it doesn't have to match the fabric on the front. But IMHO, I would use one from the front. And you don't have to place the inset fabric in the middle either. Your LA-er would probably appreciate it if you don't run it straight up and down as the seams create bulk and the quilt will probably stretch unevenly as she rolls it, creating tucks or wrinkles.
On my backs, I usually place the inset piece a little off of straight, just enough to make it look like I intented it. Here are a couple of examples you might enjoy that I found. The third one is the back I made for my Mom's quilt: