Symmetrize Texture
Install
To install this add-on, choose [edit] > [preferences...] > [Add-ons] > [Install...] > choose downloaded ZIP file.
To enable the add-on for future uses, check the box next to its name to enable it.
'Symmetrize Texture' panel will be appeared in the Image Editor and 3D View. The 3D brush only works in Object Mode. It does not work in Texture Paint mode.
To uninstall the add-on, you need to restart the blender before you do it. (Some parts of the add-on run as a shared library)
Axis Setting
This is an important setting. You have to specify which axis your image is symmetrical about.
Note that the add-on does not care if your UV map or mesh is symmetrical. Fix the UV map using the Symmetrize UV Util add-on in case it is not symmetrical.