let the topic = x
let the whole (the organising concept) = T
x is related to T thus, x:T, : changes depending on the organising relation
inverse topic is T-x
so T = x + T-x is what you have stated.
George Boole, Lincoln, 1847, The Mathematical Analysis of Logic, being an Essay towards a Calculus of Deductive Reasoning http://www.gutenberg.org/files/36884/36884-pdf.pdf See the Table on page 25
the reverse appears to me to be T′, as you say, that's a very big space
it's unclear to me what you intend by the converse. you say that it is located within T′, let's call it x′ (pronounced x prime).
following on from that the obverse appears to be T′-x′.
the whole of the topic (T) and its reverse (T′) could be 1 using this notation, and this notation is fairly standard.
Boole called the conceptual space we're trying to pin down the “domain of discourse”, this is fairly standard terminology.
> Note that I used the word "universe" just now. It turns out that there's just no good word for the combination of the whole of a topic and it's reverse.
In that case I don't think you should use the word universe for that designation. Maybe just use domain of discourse? However, be aware that there is a precedent for the use of the word universe: https://ncatlab.org/homotopytypetheory/show/universe
> kind of logic
To my mind what you are doing is not logic per se but analysis. Specifically a narrow kind of conceptual analysis https://philpapers.org/browse/conceptual-analysis Though if pressed I would not quibble with you calling what you are doing a kind of logic. Again, there is a precedent for this.
I'm kind of intrigued as to what you're getting with the notion of a converse. Maybe give a concrete example? From there it should be obvious what is meant by the obverse.