  Palindromes are popular and important objects in textual data processing,
bioinformatics, and combinatorics on words. Let $S = XaY$ be a string, where
$X$ and $Y$ are of the same length and $a$ is either a single character or the
empty string. Then, there exist two alternative definitions for palindromes:
$S$ is said to be a palindrome if: Reversal-based definition: $S$ is equal to
its reversal $S^R$; Symmetry-based definition: its left-arm $X$ is equal to the
reversal of its right-arm $Y^R$. It is clear that if the "equality" ($\approx$)
used in both definitions is exact character matching ($=$), then the two
definitions are the same. However, if we apply other string-equality criteria
$\approx$, including the complementary model for biological sequences, the
parameterized model [Baker, JCSS 1996], the order-preserving model [Kim et al.,
TCS 2014], the Cartesian-tree model [Park et al., TCS 2020], and the
palindromic-structure model [I et al., TCS 2013], then are the reversal-based
palindromes and the symmetry-based palindromes the same? To the best of our
knowledge, no previous work has considered or answered this natural question.
In this paper, we first provide answers to this question, and then present
efficient algorithms for computing all maximal generalized palindromes that
occur in a given string. After confirming that Gusfield's offline suffix-tree
based algorithm for computing maximal symmetry-based palindromes can be readily
extended to the aforementioned matching models, we show how to extend
Manacher's online algorithm for computing maximal reversal-based palindromes in
linear time for all the aforementioned matching models.
