If you want to find palindromes, why not do it with a regex directly?

#!/usr/bin/perl
use strict;
use warnings;

my $str = do {local $/; <DATA>};
$str =~ s/\s+//g;

while ( $str =~ m/( (..+) .? (??{ reverse $2 }) )/xgc )
{
   print pos( $str ) . ": $1\n";
   pos( $str ) = $-[0] + 1;      # slide pos back to the left
}
[download]

prints:

14: AGGGA
21: TACAT
25: GTTG
55: GAAAAAAAG
...etc...
[download]

I'm not sure how it would compare with the two-stage approach for speed, though. It is much faster if you minimize the qualifier ..+?, but then you end up with the shortest palindrome at each position, rather than the longest.

I make the assumption that you don't care about palindromes shorter than 4 characters. If you bump that upwards, things get faster.

Update: I tested, and it looks like the substr approach is considerably faster, particularly if you do it in a single pass.