groups of groups using self joins in DBIx::Class

I'm trying to understand how to use DBIx::Class.

If I want groups of records such that groups can themselves be members of groups, I might create a schema that includes something like this:

CREATE TABLE groups (
       id    INTEGER PRIMARY KEY,
       name  VARCHAR(100)
       );

CREATE TABLE group_groups (
       parent_id         INTEGER REFERENCES groups(id),
       child_id          INTEGER REFERENCES groups(id),
       PRIMARY KEY(parent_id,child_id)
       );

If I use DBIx::Class::Schema::Loader to dump this schema, I get the following relationships:

Group.pm

__PACKAGE__->has_many(
  "group_groups_children",
  "Schema::Result::GroupGroup",
  { "foreign.child_id" => "self.id" },
  { cascade_copy => 0, cascade_delete => 0 },
);

__PACKAGE__->has_many(
  "group_groups_parents",
  "Schema::Result::GroupGroup",
  { "foreign.parent_id" => "self.id" },
  { cascade_copy => 0, cascade_delete => 0 },
);

GroupGroup.pm

__PACKAGE__->belongs_to(
  "child",
  "Schema::Result::Group",
  { id => "child_id" },
  {
    is_deferrable => 1,
    join_type     => "LEFT",
    on_delete     => "CASCADE",
    on_update     => "CASCADE",
  },
);

__PACKAGE__->belongs_to(
  "parent",
  "Schema::Result::Group",
  { id => "parent_id" },
  {
    is_deferrable => 1,
    join_type     => "LEFT",
    on_delete     => "CASCADE",
    on_update     => "CASCADE",
  },
);

I think I need a many_to_many relationship bridge where the group table is on both sides, so I created this:

__PACKAGE__->many_to_many(
    'childgroups' => 'group_groups_children' , 'child');

__PACKAGE__->many_to_many(
    'parents' => 'group_groups_parents' , 'parent');

since my understanding of a the relationship bridge definition goes like this: 'accessor_name' => 'name of has_many relation in the related table', 'name of belongs_to relation in the relating table'

When I attempted this code:

my $group_rs = $schema->resultset('Group')->search(
        { id => $id }
);
my $group = $group_rs->first;
foreach my $child ($group->childgroups) {
<snip>

Setting DBIC_TRACE showed the SQL to be:

SELECT child.id, child.name FROM group_groups me 
       JOIN groups child ON child.id = me.child_id  
       WHERE ( me.child_id = ? )

But I think the line should look more like:

SELECT child.id, child.name FROM group_groups me 
       JOIN groups child ON child.id = me.child_id 
       JOIN groups parent ON parent.parent_id = me.id 
       WHERE ( me.child_id = ? )

If someone would suggest how I am misunderstanding the many_to_many relationship bridge and correct my many_to_many function definitions, I would be grateful.