Introduction: SUBSTANTIALITY AND METHOD
The idea here goes back to a concern of Adorno's that has come up before: the required unity of form and content. Here, form is philosophical method, and content or substantiality is what is being analyzed or philosophized about.
That generation, also Simmel, Husserl, and Scheler, sought in vain for a philosophy which, receptive to the objects, would render itself substantive. What tradition dismissed is what tradition desired. This does not obviate the methodological consideration, of how substantive particular analysis stands in relation to the theory of dialectics. The idealistic-identity philosophical avowal that the latter dissolves itself in the former is unconvincing. Objectively, however, the whole which is expressed by theory is contained within the particular to be analyzed, not first through the cognizing subject. The mediation of both is itself substantive, that through the social totality.
Adorno shares the famous goal of Husserl's:
We can absolutely not rest content with “mere words” […]. Meanings inspired only by remote, confused, inauthentic intuitions—if by any intuitions at all—are not enough: we must go back to the “things themselves”. — Husserl, Logical Investigations
Negative dialectics reflects this attitude in its scepticism towards conceptual overreach, and in its "priority of the object". But it also aims to show that phenomenology and related philosophies couldn't possibly achieve their goal, because they used the wrong methods (wrong because they relied on immediacy and thus did not appreciate mediation). Adorno is showing how his method is better than those of both (a) Bergson, Simmel, Husserl, and Scheler; and (b) Hegel. The former used flawed methods and the latter used a good method (dialectics) in a flawed manner (idealism).
The crucial statement is...
the whole which is expressed by theory is contained within the particular to be analyzed
Dialectical theory is
brought out of the particulars, because it is latent within them already — things and relations in reality are dialectical. This relationship between substance (particulars) and method (theory) is what this section is about, and in a way it's what dialectics is all about too.
More concretely what he is saying here is that the only way to get to the things themselves, to really be receptive to objects, is to see them as nodes of the social whole.
It is however also formal due to the abstract nomothetism [Gesetzmaessigkeit] of the totality itself, that of exchange. Idealism, which distilled its absolute Spirit out of this, encrypted something true at the same time, that this mediation encounters phenomena as a compulsory mechanism; this lurks behind the so-called constitution-problem. Philosophical experience does not have this universal immediately, as appearance, but as abstractly as it objectively is. It is constrained towards the exit of the particular, without forgetting what it does not have, but knows. Its path is doubled, similar to the Heraclitean one, the upwards and the downwards. While it assures itself of the real determination of the phenomena through its concept, it cannot profess this ontologically, as what is true in itself. It is fused with what is untrue, with the repressive principle, and this lessens even its epistemological dignity. It forms no positive telos in which cognition would halt. The negativity of the universal solidifies for its part the cognition into the particular as that which is to be rescued. “The only thoughts which are true are those which do not understand themselves” [Adorno quotes himself, from Minima Moralia].
Adorno points out that the abstract character of the social whole is not an immediate given in philosophical experience. In starting with the particular, and moving out from there, one starts with the appearance and moves to the abstract conceptual form. Now, the latter is ideological — remember that we do not impose our own concepts from the outside but follow the meaningful concepts that exist — and only falsely describes the particular that we started with. Or more fundamentally, the concept that identifies the particular, e.g., as a worker or a commodity, actually contains the repressive principle, so there is an essential falsity in understanding owing to the provenance of the concepts we must use.
This "negativity of the universal," which means formal, logical abstraction's denial of the particular's specificity, has the result (if we are critical and self-reflective in the way that Adorno says is necessary for philosophical experience) of fixing our thought and knowledge back onto the particular again, which is now seen as something that needs rescuing from its conceptual shackles.
I see the meaning of the MM quotation as something like this: thoughts which do not understand themselves are thoughts which are not captured by higher-order concepts or systems thereof.
On second thoughts, it's about arrogance vs. self-awareness and humility: thoughts which "understand themselves" are thoughts which are unaware of how insufficient they are.
In their inalienably general elements, all philosophy, even those with the intention of freedom, carries along the unfreedom in which that of society is prolonged. It has the compulsion in itself; however this latter alone protects it from regression into caprice. Thinking is capable of critically cognizing the compulsory character immanent to it; its own inner compulsion is the medium of its emancipation.
This follows naturally from what went before. All philosophy contains and tends to perpetuate the repression or domination at the heart of society, because its concepts are sociohistorical sedimentations of repressive social relations. To anticipate: this is why, since we must do philosophy, we have to do it critically, so as to minimize the perpetuation.
Then Adorno says that this very feature, namely that it contains unfreedom, is what allows philosophy to be self-critical and prevent the perpetuation of unfreedom even though it remains within it.
But how can it be both disease and cure at the same time? My vague and intuitive first attempt is: the other aspect to this compulsory character of thought is its compulsive pursuit of understanding. It doesn't surrender itself to "caprice," in other words, it has the strength to follow through to the end, to try to capture the particulars.
To be slightly less vague: as I said before, social unfreedom is carried into philosophy in the form of abstraction, formalism, and logic. This logic is
rigorous, and it is this compulsion of rigour that allows thought to push through to the truth. Philosophy doesn't find the truth in the fantasy that it can cast aside all restraints and do what it wants — this is mere "caprice" — but by knowingly working within the confines of the abstractions whose sources it aims to question.
The freedom towards the object, which in Hegel resulted in the disempowerment of the subject, is first of all to be established. Until then, dialectics diverges as method and as one of the thing. Concept and reality are of the same contradictory essence. What tears society apart antagonistically, the dominating principle, is the same thing which, intellectualized, causes the difference between the concept and that which is subordinated under it.
It follows from what has gone before that antagonistic reality and antagonistic method/theory are two sides of the same coin. Or rather, they are the same thing, the latter just being the "intellectualized" aspect of the former. Specifically, the antagonism in society finds its expression in theory in the antagonism and divergence "between the concept and that which is subordinated under it." And this latter appears as contradiction.
So Adorno then repeats the central points he made earlier in the introduction:
The logical form of the contradiction however achieves that difference, because every one which does not suborn itself to the unity of the dominating principle, according to the measure of the principle, does not appear as a polyvalence which is indifferent to this, but as an infraction against logic.
The earlier statements to this effect were as follows:
everything which does not conform to such, everything qualitatively divergent assumes the signature of the contradiction. The contradiction is the non-identical under the aspect of identity; the primacy of the principle of contradiction in dialectics measures what is heterogenous in unitary thinking. By colliding against its own borders, it reaches beyond itself. — Dialectics Not a Standpoint
Contradiction is non-identity under the bane of the law — Dialectics Not a Standpoint
Moving on...
On the other hand the remainder of the divergence between philosophic conception and follow-through also testifies to something of the non-identity, which neither permits the method to wholly absorb the contents, in which alone they are supposed to be, nor intellectualizes the contents. The pre-eminence of content reveals itself as the necessary insufficiency of the method.
It feels like he is trying to turn the negative into a positive, in emphasizing that the divergence has critical potential and contains a path to truth.
Between philosophical conception and follow-through (execution) there is a divergence because of the divergence between concept and object already described. But in the execution there is a remainder, which I think is either a receptivity to the non-identical, or is just the non-identical itself (which agrees with your interpretation
@Metaphysician Undercover).
Another way to put that is that Adorno is moving from a description of the divergence between concept and object to an emphasis that in philosophical experience, particularly the execution of dialectical method, this divergence has a substantive remainder, namely the non-identical itself. That is, this gap between concept and object isn't just empty.
What as such, in the form of general reflection, must be said, in order not to be defenseless against the philosophy of the philosophers, legitimates itself solely in the follow-through, and is negated therein in turn as method. Its surplus is with respect to its content abstract, false; Hegel already had to accept this discrepancy in the preface to the Phenomenology. The philosophical ideal would be to render the accounting one would give for what one does superfluous, by doing it.
Note that the surplus and the remainder are not the same thing. The surplus is that of method and theory; the remainder is non-identical content.
The surplus is "the accounting one would give for what one does," i.e., the conceptual superstructure. It is in the methodological execution that thought moves to the particulars and leaves the methodology behind, whereupon you find the non-identical remainder, and (ideally) cast aside the theoretical baggage, i.e., the surplus.