These are the categories, the x marks the categories a random post has selected.
=============
[ ] Movies
-- [ x ] Comedy
-- [ x ] Thriller
-- -- [ x ] Noir

[] Music
-- [] Dance
============

When I run
wp_get_object_terms($post->ID, 'project_category', array('orderby' => 'term_order'));

The result is below. As you can see the Thriller is first instead of the expect Comedy.

Array
(
    [0] => stdClass Object
        (
            [term_id] => 32
            [name] => Thriller
            [slug] => thriller
            [term_group] => 0
            [term_taxonomy_id] => 35
            [taxonomy] => project_category
            [description] =>
            [parent] => 31
            [count] => 1
        )

    [1] => stdClass Object
        (
            [term_id] => 33
            [name] => Noir
            [slug] => noir
            [term_group] => 0
            [term_taxonomy_id] => 36
            [taxonomy] => project_category
            [description] =>
            [parent] => 32
            [count] => 2
        )

    [2] => stdClass Object
        (
            [term_id] => 34
            [name] => Comedy
            [slug] => comedy
            [term_group] => 0
            [term_taxonomy_id] => 37
            [taxonomy] => project_category
            [description] =>
            [parent] => 31
            [count] => 1
        )

)