As preface to this post, there are two prior posts which are:
[outlining MBTI correlations to the enneagram, IMHO]
As noted previously [the second post there], I’m not particularly a believer in ‘tritype’ theory as its envisaged at the moment. It is however a topic that’s at least conceptually interesting. People theoretically have multiple centers, so its not unreasonable for them to have multiple preferences in different ‘parts’ of each centre, which would then somewhat to a second ‘type’ preference in that centre.
The key question, as noted previously, is that some enneagram theories predict centers other than the main centre are opposed to it. For instance, Hurley-Donson’s “repressed center” theory is somewhat at odds with it, or at least some combinations of it. They allocate 459 as missing ‘enactive’, 379 as missing ‘feeling’ and 126 as missing ‘thinking’ centre – consequently that goes against the idea of an 8 having a 234 subtype (except perhaps 3).
This gets more complicated if we assume that a type has problems not with just a whole centre, but with part of a centre. Jungian functions for example can stand in for particular ‘parts of centres’; one part of centre is a ‘counter-type’ of the main type and is fixed for each type, while wing probably determines the other numbers.
This doesn’t invalidate tritype exactly, so much as as there is a ‘predicted’ tritype which equates to the normal function organization for a given type/wing. For example, if someone has secondary introverted feeling, that probably equates to (or explains) a tritype result of ‘4’. Extraverted feeling on the other hand, would be a tritype of ‘2’.
Overall using the initial correspondences we get enneatype correspondences by functions as:
1w9 [Te/Si/Ne/Fi] – [1,9] + 7, 4
1w2 [Te/Ni/Se/Fi] – + 6, 8, 4
2w1 [Fe/Si/Ne/Ti] +9,7,5
2w3 [Fe/Ni/Se/Ti] – + 6, , 5
3w2 [Se/Ti/Fe/Ni] – 3+ 5, , 6
3w4 [Se/Fi/Te/Ni – [3,4] + 1,6
4w3 [Fi/Se/Ni/Te] – [4,3] +6,1
4w5 [Fi/Ne/Si/Te] –  + 7,9,1
5w4 [Ti/Ne/Si/Fe] –  + 7,9,2
5w6 [Ti/Se/Ni/Fe]  + 8, , 2
6w5 [ Ni/Fe/Ti/Se] –  +2, , 3*
6w7 [Ni/Te/Fi/Se] –  + 1,4, 3*
7w6 [Ne/Ti/Fe/Si] –  + 5,2,9
7w8 [Ne/Fi/Te/Si] –  + 4,1,9
8w7 [Se/Ti/Fe/Ni] –  + 5,2,6
8w9 [Se/Fi/Te/Ni] –  + 5,2,6
9w8 [Si/Fe/Ti/Ne] –  + 2,5,6
9w1 [Si/Te/Fi/Ne] –  + 1, 4, 7
[* I have allocated Se as 3 for type 6, otherwise 8 ; on the theory this would line up with the ‘arrows’]
This does run into some problems e.g. as One = Te, and Five = Ti, a couple of types can have two lead functions in what seem to be the same ‘centre’.
The function setups here may not always coincide with people’s self-evaluated tritype arrived at independently of this, since a lot of traits associated with a type are not necessarily to do with its function. For example, a 5w4 has an ‘Ne’ auxiliary here, which functionally is similar to 7, but the function itself is just about in a sense ‘applied imagination’. A 7s other traits like avoidance of pain and fun-seeking are less present, and many 7 traits are considerably different to the 4 traits generated by the wing, making it harder to find. Or, a 5w6 Se auxiliary is ‘8-like’, but deals with information processing only, not so much how eights’ interact with people.
Note also that a ‘type’ viewed as a dominant function, has some ‘unhealthy’ traits which are actually generated by its particular inferior function. Where a function is secondary, unhealthy traits are the third function, and much less discernable. In other words, in a function model the upside and downside of the type are ‘sold separately’. An 8 for example with Ti secondary has a degree of ‘Five-ish’ thinking abilities, without necessarily having much of the 5s troubles with too much introversion or problems with relationships (since Fe is 3rd rather than 4th). Tritype is saying only that ‘this is present’, but it works differently from Jung in that Jung defines the ‘order’ of functions, so the last couple of functions are underdeveloped or poor/inferior. We can ask for instance if a 5 has trouble with interpersonal relationships [Fe inferior] , whether we should keep the final ‘2’ in the tritype or not – i.e. would a 5 regard themselves as having a 2 in their tritype, if they were asked? Maybe not.