Based on Math behind Colourspace Conversions, I was able to at least sort the colours:

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

sub hue {
    my ($colour) = @_;
    my %rgb;
    @rgb{qw{ red green blue }} = map hex($_) / 255, $colour =~ /^#(..)
+(..)(..)/;
    my @order = sort { $rgb{$a} <=> $rgb{$b} } keys %rgb;
    my $hue = {red   => sub { 0 + $rgb{green} - $rgb{blue}},
               green => sub { 2 + $rgb{blue}  - $rgb{red} },
               blue  => sub { 4 + $rgb{red}   - $rgb{green} }
    }->{ $order[2] }->();
    my $d = $rgb{ $order[2] } - $rgb{ $order[0]};
    return 0 if 0 == $d;

    $hue /= $d;
    $hue *= 60;
    $hue += 360 if $hue < 0;
    return $hue
}

my @colours = do { no warnings 'qw';
                   qw( #000000 #716373 #704A2B #AF7E45 #963049 #AA2261
                       #B24551 #E6212E #FF0000 #001200 #FFDE72 #F55B73
+ ) };
my %hue = map { $_ => hue($_) } @colours;
my @sorted = sort { $hue{$a} <=> $hue{$b} } keys %hue;
say "@sorted";
[download]

The black/white problem is not solved here, so feel free to expand as you need.