What is the fastest known game that ends that in a mutual stalemate position that is symmetrical?
[Title "Evgeny Ghik, Shakhmatyi i Matematika 1983, Non-Unique Proof Game In 18.5 Moves"] [FEN ""] 1. e4 d5 2. e5 d4 3. c3 f6 4. Qf3 Kf7 5. Qxb7 Qd5 6. Kd1 Qxg2 7. Kc2 Qxf1 8. Qxc8 Qxg1 9. Qxb8 Rxb8 10. Rxg1 Rb3 11. Rg6 Ra3 12. Rh6 gxh6 13. bxa3 Kg7 14. Kb2 d3 15. e6 a5 16. h4 a4 17. h5 c5 18. f4 c4 19. f5
In case the PDF link goes cold, here are the relevant sections.
There was a previous stack exchange answer which has a very amusing position where neither side can make a legal move.
Take a look; it is very enjoyable!