  A palindromic periodicity is a factor of an infinite word $(ps)^\omega$ where
$p$ and $s$ are palindromes and the factor has length at least $|ps|$, for
example, $accabaccab$. In this paper we describe several ways in which a
palindromic periodicity may arise through the interaction of palindromes and
periodicity, the simplest case being when a palindrome is itself periodic. We
then consider what happens when a word is a palindromic periodicity in two
ways, a situation similar to that considered in the Fine and Wilf Lemma and
obtain something slightly stronger than that lemma. The paper ends with
suggestions for further work.
