Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns.
A few decorative examples do however exist; in base 10 the following are primes: 11, 122333221, and 1223334444555554444333221.
It was found on 18 October 2021 by Ryan Propper and Serge Batalov.
All binary palindromic primes except binary 11 (decimal 3) have an odd number of digits; those palindromes with an even number of digits are divisible by 3.
Belphegor's Prime consists of the number 666, on either side enclosed by thirteen zeroes and a one.