Your main issue is in the lines

    $groups{$_}{left}   = $rxn[0] for @left_els;
    $groups{$_}{right}  = $rxn[0] for @right_els;
[download]

If a group value turns up more than once, you overwrite the key value that you stored before. I am lazy, so here is a script that I think does what you want (rather than fixing yours). I tried to keep it as close as possible to yours so you can take parts and paste it into yours.

use strict;
use warnings;

my %groups;
#my @keys;
while(<DATA>){
        s/(\s|#)\d\s/$1/g;
        s/\s//g;
        my( $key, $left, $right ) = /^([^#]+).*#(.*)<=>(.*)/;
        $groups{$_}{left }{$key} = 1 for split /\+/, $left;
        $groups{$_}{right}{$key} = 1 for split /\+/, $right;
        #push @keys, $key;
}

for my $grp ( values %groups ) {
        for my $left ( keys %{ $grp->{left} } ) {
                for my $right ( keys %{ $grp->{right} } ) {
                        print "$left <=> $right\n";
                }
        }
}
__DATA__
R00009MM#R00009#2 C00027MM <=> C00007MM + 2 C00001MM
R00014MM#R00014#C00022MM + C00068MM + C00080MM <=> C05125MM + C00011MM
R00081MM#R00081#C00007MM + 4 C00126MM + 8 C00080MM <=> 4 C00125MM + 2 
+C00001MM + 4 C00080Cyto
R00086MM#R00086#C00008MM + C00009MM + 7 C00080Cyto <=> C00002MM + C000
+01MM + 7 C00080MM
R00094MM#R00094#2 C00051MM + C00003MM <=> C00127MM + C00004MM + C00080
+MM
[download]

UPDATE: Just realized that @keys is not used, so I commented those two lines.