What you might want to do, bearing in mind the structure of a palindrome, is use two loops to iterate over each half of the palindrome, checking If the letters at the opposite indexes match. So take [r,a,c,e,c,a,r]. Index 0 matches index 6 (r), index 1 matches index 5.
You might not even need two loops, just think about the maths involved in finding where two matching letters are in relation to eachother