I think what Chloee is talking about is the ordering of disintegration and integration that runs a line of 1-4-2-8-5-7 and 3-9-6 or 1-7-5-8-2-4 and 3-6-9. When you look at the type descriptions, along with their component desires and fears, you can see how the directions of disintegration and integration describe what similarities certain types have to other types with regard to being unhealthy/healthy and stressed/confident.
Chloee may also be talking about the triads. The three intelligence centers (gut, image, head) from a triad (1,9,8-2,3,4-5,6,7), the Hornevian groups (withdrawn, assertive, compliant) form a triad (5,4,9-7,8,3-1,2,6), the coping styles (reactive, positive outlook, competency) form a triad (4,6,8-9,7,2-1,3,5), and the expressive style (out of touch with, underexpressive of, overexpressive of...primary intelligcene center) form a triad (3,6,9-5,2,8-1,4,7). These four triads arrange the types in all possible configurations, explaining one key quality that each type has in common with two other types.
The arrangment in order along the enneagram symbol results from the intelligence center triad being the most primary triad...it's the one that's most inborn, ingrained, it has more to do with the way your brain is wired. From here, the next traid in the heirarchy is how one responds to the intelligence center. One can either overexpress, underexpress, or be out of touch with one's primary intelligece center. The grouping of these two most fundamental traids account for type ordering along the symbol, explaining why wings must be adjacent to core type. If you arrange the traids differently, making, say, the coping styles most primary, then you really mix things up because the intelligence centers are most primary and coping styles derive ultimately from that...if you arrange wings by coping style then you are arranging them by something more derivative and less fundamental...they woudln't even really be a wing, be definition, anymore. If you arrange wings randomly, stating that any type can have any wing, then you are ignoring the fundamental dynamics of the system as just explained, and it would not longer matter that a fixation starts from the fundamental aspect of intelligence center, then one's reaction to it, then how one copes with it, etc.