It would be cool to reuse this regexp search for the forward/backward test, but I don't know how resp. if this is even possible.

If you drop your "without reverse" requirement, you can use it in the regex!

/(^.*\z)(?(?{ $+ eq reverse $+ })|(?!))/s
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Or, in a more verbose format:

/
    (                            # capture group
        ^                        # match begin of string
        .*                       # match everything
        \z                       # match end of string
    )
    (?                           # if
        (?{ $+ eq reverse $+ })  # $+ (last captured group) is palindr
+ome
                                 # then match nothing (succeed)
    |                            # else
        (?!)                     # fail on matching nothing (fail the 
+regex)
    )
/xs                              # x for readability, s to make . matc
+h \n

# Note: can match empty string
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Depending on what you want to match and whether you want backtracking, change the contents of the capture group.

I think I prefer Perl 6 regexes here :)

rule palindrome {
    $foo := (\w+) ::: { fail if $foo ne reverse $foo }
}

# Note: doesn't match empty string
[download]

(assuming a \w+ word and no backtracking)

Or a simpler version, but this always backtracks:

/(.+)(??{ reverse $+ })/s
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Explained:

/
    (.+)                          # match one or more characters
    (??{ quotemeta reverse $+ })  # match the reverse of what was matc
+hed
/xs

# Note: doesn't match empty string or single character string
[download]

A Perl 6 version:

rule palindrome {
    $foo := (\w+) <{ quotemeta reverse $foo }>
}
[download]