A word is called a *palindrome* if it is equal to its reversal. In the paper we consider a *k*-abelian modification of this notion. Two words are called *k*-*abelian equivalent* if they contain the same number of occurrences of each factor of length at most *k*. We say that a word is a *k*-*abelian palindrome* if it is *k*-abelian equivalent to its reversal. A question we deal with is the following: how many distinct palindromes can a word contain? It is well known that a word of length *n* can contain at most n+1

distinct palindromes as its factors; such words are called *rich*. On the other hand, there exist infinite words containing only finitely many distinct palindromes as their factors; such words are called *poor*. We show that in the *k*-abelian case there exist infinite words containing finitely many distinct *k*-abelian palindromic factors. For rich words we show that there exist finite words of length *n* containing Θ(n2)

distinct *k*-abelian palindromes as their factors.