Do Geese see God?
Never odd or even.
They can, of course, theoretically exist in any language. There is a famous known Palindrome from ancient Rome called the Sator Square. It's attested as early as 70AD in Pompeii and scattered throughout the Empire:
It reads in the square right, left, up, and down as:
SATOR AREPO TENET OPERA ROTAS
THE SOWER AREPO HAS WORKS (IN) WHEELS
In my novel, Amor Vincit Omnia: an Andrew Valquist Adventure, the hero is a newly hired employee at the NSA (a matter on which I have some expertise). On his second day of orientation, however, he's whisked away to a mission in Romania where a knowledge of Latin is key to success. He then is put in charge of a team tasked with translating a Latin book which somehow cryptically holds a secret message about an upcoming attack. Just when they think they've figured everything out, an unexpected Latin palindrome emerges! Here's a hint. Amor (love) is a palindrome for Roma (Rome).