I put some thought to the palindrome writing system. Or, at least one of them.
There are a few different writing systems used by palindromes, but by far the most widespread one is this "reverse abugida" developed by grassland palindromes some time ago.
(an abugida is a writing system that uses mostly consonants, and then uses some kind of other notation to imply what kind of vowel is attached to said consonant. like a small symbol on the consonant, or changing the orientation of the consonant, and so on. Here, it's the other way around, with vowels being the base and consonants being the additive thing.)
Vowels use large lines that connect together.
Consonants are smaller symbols like dots, circles or dashes that punctuate those lines.
Palindromes don't have access to as many consonants as we have because air doesn't travel between their palate and tongue, but instead over/under both of them. So, they don't get to obstruct air coming out of them in the same ways that we do.
Put them together, and you get words!
Now for the hard part....... to figure out *what* those words are.