There's an easier way to do that. You don't have to create a new layer at all.
1. Select region
2. Ctrl-C
3. Ctrl-V
4. Shift F and click on the floating selection
5. M and move it appropriately, then click outside the selection, and it will automatically anchor. It's way faster.
If you want to do this often, put a guide in the very center of the image so that your mirrored image will snap to the center when you move it.