Try this:

my $bitfield = 1;
my $modifier = '+2+16+1024';

my @fields = ( $modifier =~ m{ ([+-])  (\d+) }gxms ); # extract mods
while ( @fields >= 2 ) {                # for each modification
    my $op    = shift @fields;
    my $value = shift @fields;

    if ( $op eq '+') {
        $bitfield |= $value;            # set a bit w/ OR
    }
}
print "bf: $bitfield\n";
__END__
bf: 1043
[download]

That's the desired result, right? If I change the first line:

my $bitfield = '1';
[download]

...then I get 3624. This isn't the result you got (3562), but I still think the problem is that you're starting out with a string somewhere, or you're using strings where you think you have numbers. This way...

my $bitfield = '1';  $bitfield = 0+$bitfield;
[download]

...the result goes back to 1043.