Interesting…
Are you now; closing the Symmetry tool, hitting Save and then closing the painting without using any other tools?
If those exact steps, then yes the next time the painting is opened for me the Symmetry tool is again automatically active.
To get out of this cycle some other action that can be “Saved” within the painting needs to happen in addition to just closing the Symmetry tool.
Try adding a new layer, or making a mark or something after closing the Symmetry tool and then hit Save and close the painting.
Hopefully the next time the painting is opened the Symmetry tool will not automatically be there again.