opendir my $DIR, '.' or die "Can't opendir . : $!\n";
my %single;
my %other = (did => 'mrg', mrg => 'did');

while (my $file = readdir $DIR) {
#print "$file\n";
    my ($name,$ext) = ($file =~ /^(.*)\.(did|mrg)$/)
        or next;
    if (exists $single{"$name.$other{$ext}"}) {
        delete $single{"$name.$other{$ext}"};
    } else {
        $single{$file} = undef;
    }
}

print "Unpaired files:\n";
foreach my $file (keys %single) {
    print "\t$file\n";
}
[download]

A bit longer than most but if the readdir gives you the filenames in order (my does), there is a lot of such files and only a minimum unpaired I believe it would be the quickest and least memory hungry.

Update: Of course if your filesystem is case insensitive you should lowercase the filename to make sure you look for the pairs case insensitively. You may want store the original case in the %single hash in such case.

Edit by castaway: Closed small tag in signature