In the Edit Menu, there are some options like:
Mirror Paste Vertically
Mirror Paste Horizontally
Although I don't know why they are disabled sometimes. You could try and use those to help you achieve symmetry.
EDIT: Sorry, I finally found out how to use it.
First, you copy and paste a part of the map. Then, you use the Mirror Paste Vertically thing to flip the paste. This way, you can copy half of the map, use these things and flip the paste, and paste it on the map to achieve symmetry.