Reinin dichotomies were derived using group theory, a field of math I know very little about. This site gives a thorough explanation. Unfortunately it is in Russian, and online translators obviously don't do very well with the mathematical terminology. That said, maybe misutii and any resident mathematicians could help out.
I'll try to explain my current understanding of it as simply as possible. The idea is that the socion, with its dichotomies, is a Klein 4-group. This means, given a few initial dichotomies, you can produce all other possible symmetric dichotomies. (I'm not sure what symmetric means in this context.)
New dichotomies can be formed by combining the original four dichotomies. If I, N, T, and J are represented by 1 (true) and their opposites by 0 (false), comparing them with the boolean operation of equivalence (=) yields new dichotomies. For instance, a type is "static" if I and J are both true or both false. Mathematically, I=J. (The above article uses the clumsy but equivalent notation (I & J) v (~I & ~J) instead.) Any other commutative (i.e. symmetric) operation, such as XOR, could also be used.
4C2 = 6 dichotomies are formed by comparing pairs of the original four dichotomies.
(Note that the resulting names are all applicable to INTj, because of the way we chose our truth values at the beginning. Unfortunately there are no neutral names for each dichotomy; one is forced to say Static-Dynamic. However, it's easy to do this based on any other type.)
These new second-order dichotomies can be compared to create even more dichotomies.
reasonable: static=N : (I=J)=N
subjectivist: static=T : (I=J)=T
However, the number of distinct combinations is limited by the properties of equivalence. (I=J)=T is the same as (T=I)=J, because equivalence is associative--that is to say, the order doesn't matter--as well as commutative. Given that, you might as well just get rid of the equals signs and parentheses, and write the original dichotomies that each new dichotomy depends on.
Including the original dichotomies, for completeness:
There is but a single fourth-order dichotomy: Questioner-Declarer. I have heard that Aushra identified this one herself. Coincidence?
Except for the first- and second-order dichotomies, it is a bit difficult to use these definitions in practice. Scroll to the bottom to see a more useful list, adapted from anndelise's summary.
My result shows a constraint on what kind of dichotomies are allowed: it must either always switch when one of the original dichotomies is switched, or be completely independent of that dichotomy. For example, changing J/P always changes Static-Dynamic, but changing N/S never does.
Since a dichotomy can depend on 1, 2, 3, or 4 of the original dichotomies,
all in all we have
4C1 + 4C2 + 4C3 + 4C4 = 15 dichotomies.
Remember Pascal's Triangle? sum(nCr over r=0...n) = 2[sup:52435fcc25]n[/sup:52435fcc25]. Therefore, since we exclude the null dichotomy (which isn't really a dichotomy at all) there must be 2[sup:52435fcc25]4[/sup:52435fcc25] - 1 dichotomies in all; 4 original and 11 Reinin. I suppose there is also a group-theoretical explanation for this. An interesting consequence is that in any dichotomy-based type theory the total number of dichotomies is 1 less than the number of types!
I mentioned before that every pair of non-identical types shares exactly 7 dichotomies. Why 7? (This is another question for our imaginary mathematician.)
Since Reinin dichotomies treat the socion symmetrically, there is a slight mismatch between functional and dichotomal interpretations. For instance, in asymmetrical relations, sometimes one's Supervisor shares the Reasonable dichotomy, and sometimes not. However, all the symmetrical relationships are consistent in this respect. When I can, I will post a table showing the dichotomy changes for each relationship. I think the ones for Duality, Conflict, and the other intra-quadra relations may be especially revealing.
farsighted = IN + ES
careless/happy-go-lucky = EN + IS
obstinate = IT + EF
compliant = ET + IF
static = Ij + Ep
dynamic = Ip + Ej
democratism = Q1 + Q3 = NT + SF
aristocrats = Q2 + Q4 = ST + NF
tactical = Np + Sj
strategic = Sp + Nj
constructivist = Fj + Tp
emotivist = Tj + Fp
process/rightist = PQ1 + JQ2 + PQ3 + JQ4 = NTp + SFp + STj + NFj
result/leftist = JQ1 + PQ2 + JQ3 + PQ4 = NTj + SFj + STp + NFp
cheerful/gay/subjectivism: Q1 + Q2 = ITj + EFj + IFp + ETp
serious/objectivist: Q3 + Q4 = ITp + EFp + IFj + ETj
reasonable = Q1 + Q4 = INj + ESj + ISp + ENp
resolute/decisive = Q2 + Q3 = INp + ESp + ISj + ENj
positivist = EQ1 + IQ2 + EQ3 + IQ4 = ENT + ESF + IST + INF
negativist = IQ1 + EQ2 + IQ3 + EQ4 = INT + ISF + EST + ENF
asker/taciturn = NQ1 + NQ2 + SQ3 + SQ4 = TQ1 + FQ2 + FQ3 + TQ4
declarer/narrative = SQ1 + SQ2 + NQ3 + NQ4 = FQ1 + TQ2 + TQ3 + FQ4