Palindromic primitives and palindromic bases in the free group of rank two

The present paper records more details of the relationship between primitive elements and palindromes in F₂, the free group of rank two. We characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F₂. Further, we note that each cyclically reduced primitive element is either a palindrome, or the concatenation of two palindromes.