We consider the problem of characterizing the palindromic sequences 〈cd−1, cd−2, . . . , c0〉, cd−1 6= 0, having the property that for any K ∈ N there exists a number that is a palindrome simultaneously in K different bases, with 〈cd−1, cd−2, . . . , c0〉 being its digit sequence in one of those bases. Since each number is trivially a palindrome in all bases greater than itself, we impose the restriction that only palindromes with at least two digits are taken into account. We further consider a related problem, where we count only palindromes with a fixed number of digits (that is, d). The first problem turns out not to be very hard; we show that all the palindromic sequences have the required property, even with the additional point that we can actually restrict the counted palindromes to have at least d digits. The second one is quite tougher; we show that all the palindromic sequences of length d = 3 have the required property (and the same holds for d = 2, based on some earlier results), while for larger values of d we present some arguments showing that this tendency is quite likely to change. Mathematics Subject Classification (2010): 11A63