The following is slow reading and I'm aware of that... sorry that I couldn't revise it to be any less tiring!

Supervision comprises a group, in a loose sense...
- Your supervisor's supervisor is your superego.
- Your superego's supervisor is your supervisee.

As does the beneficiary relation...
- Your beneficiary's beneficiary is your superego.
- Your superego's beneficiary is your benefactor.

With each group containing four elements, and two in the intersection.

Then, what if we take these two groups and combine them, such that it is possible to construct a path between any two people exclusively through the use of supervisor/supervisee and benefactor/beneficiary relations?

We find that the dual is the supervisor's beneficiary and the contrary is the supervisor's benefactor. Surprisingly, the dual is also the beneficiary's supervisor and the contrary is the beneficiary's supervisee.

The lattice traversed by the two relations contains eight types, no matter which type you start: Identical, dual, supervisor, supervisee, benefactor, beneficiary, superego, contrary. It has the following pattern:
[table:a351a15b01]
[row:a351a15b01]INTP[col:a351a15b01]INFJ[col:a351a15b01]ISFP[col:a351a15b01]ISTJ
[row:a351a15b01]ESTJ[col:a351a15b01]ENTP[col:a351a15b01]ENFJ[col:a351a15b01]ESFP
[row:a351a15b01]ISFP[col:a351a15b01]ISTJ[col:a351a15b01]INTP[col:a351a15b01]INFJ
[row:a351a15b01]ENFJ[col:a351a15b01]ESFP[col:a351a15b01]ESTJ[col:a351a15b01]ENTP
[/table:a351a15b01]
It contains exactly half of the types in each quadra, temperament, Reinin dichotomy, Gulenko communication group, and Gulenko erotic attitude group. And of course, half of the Socion. It is a tangible representation of the relations that take advantage of social and economic power.

Note in particular that you and your dual are on an "equal plane" in that the type that has supervision power over you also has benefactor power over your dual. Whereas your contrary has supervision power over your benefactor (two levels above)... and also is the beneficiary of your supervisee (two levels below). No wonder the two can't get along, they can't decide how to properly order things.

On the other hand, I have seen people mentioning some sort of generally unaccepted theory that contraries make good relationship partners. Perhaps just being on the same supervisor/benefactor lattice leads to an interesting interpersonal relationship with all sorts of excitement?

Anyways, what I suggest you take out of this is the following hypothesis: Dual and contrary relations might be seen as two-stage asymmetric relations... with the former "balanced" and the latter not.

For your reference, the other group is:
[table:a351a15b01]
[row:a351a15b01]ESFJ[col:a351a15b01]ENFP[col:a351a15b01]ENTJ[col:a351a15b01]ESTP
[row:a351a15b01]ISTP[col:a351a15b01]ISFJ[col:a351a15b01]INFP[col:a351a15b01]INTJ
[row:a351a15b01]ENTJ[col:a351a15b01]ESTP[col:a351a15b01]ESFJ[col:a351a15b01]ENFP
[row:a351a15b01]INFP[col:a351a15b01]INTJ[col:a351a15b01]ISTP[col:a351a15b01]ISFJ
[/table:a351a15b01]
And it contains all the types that you have a symmetric relation with: Activity, mirror, look-a-like, illusionary, comparative, semi-dual, quasi-identical, and conflicting.