A palindromic periodicity is a factor of an infinite word $(ps)^ω$ 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.