There are issues (mostly common but some not) using a regexp like this:
^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1$
with GNU grep and GNU sed, hence my contacting both mailing lists but apologies if that was the wrong starting point.

This started out as a question on StackOverflow, (https://stackoverflow.com/questions/77820540/searching-palindromes-with-grep-e-egrep/77861446?noredirect=1#comment137299746_77861446) but my "answer" and some comments from there copied below so you don't have to look anywhere else for a description of the issues.

Given this input file:
a
ab
abba
abcdef
abcba
zufolo

Removing the `$` from the end of the regexp (i.e. making it less restrictive) produces fewer matches, which is the opposite of what it should do:

a) With the `$` at the end of the regexp:

    $ grep -E '^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1$' sample
    a
    abba
    abcba
    zufolo

b) Without the `$` at the end of the regexp:

    $ grep -E '^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1' sample
    a
    abba
    abcba

It's not just GNU grep that behaves strangely, GNU sed has the same behavior from the question when just matching with `sed -nE '/.../p' sample` as GNU `grep` does AND sed behaves differently if we're just doing a match vs if we're doing a match + replace.

For example here's `sed` doing a match+replacement and behaving the same way as `grep` above:

a) With the `$` at the end of the regexp:

    $ sed -nE 's/^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1$/&/p' sample
    a
    abba
    abcba
    zufolo

b) Without the `$` at the end of the regexp:

    $ sed -nE 's/^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1/&/p' sample
    a
    abba
    abcba

but here's sed just doing a match and behaving differently from any of the above:

a) With the `$` at the end of the regexp (note the extra `ab` in the output):

    $ sed -nE '/^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1$/p' sample
    a
    ab
    abba
    abcba
    zufolo

b) Without the `$` at the end of the regexp (note the extra `ab` and `abcdef` in  the output):

    $ sed -nE '/^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1/p' sample
    a
    ab
    abba
    abcdef
    abcba
    zufolo

Also interestingly this:

    $ sed -nE 's/^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1$/<&>/p' sample

outputs:

    <a>
    <abba>
    <abcba>
    <>zufolo

the last line of which means the regexp is apparently matching the start of the line and ignoring the `$` end-of-string metachar present in the regexp! 

The odd behavior isn't just associated with using `-E`, though, if I remove `-E` and just use [POSIX compliant BREs](https://pubs.opengroup.org/onlinepubs/9699919799/basedefs/V1_chap09.html#tag_09_03) then:

a) With the `$` at the end of the regexp:

    $ grep '^\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\).\{0,1\}\5\4\3\2\1$' sample
    a
    abba
    abcba
    zufolo

<p>

    $ sed -n 's/^\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\).\{0,1\}\5\4\3\2\1$/&/p' sample
    a
    abba
    abcba
    zufolo

b) Without the `$` at the end of the regexp:

    $ grep '^\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\).\{0,1\}\5\4\3\2\1' sample
    a
    abba
    abcba

<p>

    $ sed -n 's/^\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\).\{0,1\}\5\4\3\2\1/&/p' sample
    a
    abba
    abcba

and again just doing a match in sed below behaves differently from the sed match+replacements above:

a) With the `$` at the end of the regexp:

    $ sed -n '/^\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\).\{0,1\}\5\4\3\2\1$/p' sample
    a
    ab
    abba
    abcba
    zufolo

b) Without the `$` at the end of the regexp:

    $ sed -n '/^\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\)\(.\{0,1\}\).\{0,1\}\5\4\3\2\1/p' sample
    a
    ab
    abba
    abcdef
    abcba
    zufolo

The above shows that, given the same regexp, sed is apparently matching different strings depending on whether it's doing a substitution or not.

These are the version I was using when testing above:

    $ grep --version | head -1
    grep (GNU grep) 3.11

    $ sed --version | head -1
    sed (GNU sed) 4.9

It was later pointed out that grep in git-=bash produces an error message and core dumps given the original regexp above, e.g.

    grep -E '^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1' sample

and

    grep -E '^(.?)(.?)(.?)(.?)(.?).?\5\4\3\2\1$' sample

both output:

    a
    assertion "num >= 0" failed: file "regexec.c", line 1394, function: pop_fail_stack
                                                                                      Aborted (core dumped).

Sorry, I can't copy the core off that machine for corporate reasons.

Those git-bash tests were using

    $ echo $BASH_VERSION
    5.2.15(1)-release

    $ grep --version
    grep (GNU grep) 3.0

Regards,

	Ed Morton