Let's consider a view of the dialectic as a form, or family of forms, which processes can take.
Hegel was vehemently opposed to any categorical separation of form and content. But this was one thing pushing him towards idealism, via the proposition that thought must be its own subject. So Marxists have rightly allowed some space here where Hegel saw none. John Rees quotes Lenin as favoring the "unity of knowing and being" over Hegel's "identity of knowing and being" for this reason. Form is ultimately dependent on content, "in substance and in structure", but it can be analytically isolated. Interpreting the dialectic as an abstract form rather than a property of the world adds a certain separation, but it does not entail any sort of dualism of substance, so I think we are still on safe ground.
So let us continue. The first thing we have to do is distinguish two meanings of "form" - form as appearance, as the guise in which something shows itself to us, and form as pattern, as a real structure that things can be found to have. The position we are considering here is that the dialectic is the latter, a pattern - but the two are related, and we will come back to form-as-appearance later.
Bertell Ollman provides a clear definition of the "negation of the negation" as an instance of this sort of pattern: "the process by which the most recent phase in a development that has gone through at least three phases will display important similarities with what existed in the phase before last."
This shows a strength of this interpretation of dialectics: it is readily apparent, finally, what saying a process is an instance of a dialectical pattern, in this case the negation of the negation, does and does not imply. Compare with "positive feedback", a pattern of a similar type - an example of which is the tendency of global warming to melt snow and therefore reduce the Earth's albedo and increase the amount of sunlight it absorbs. Just as once we find a positive feedback, we can say that a system will tend towards instability, with small changes being amplified, once we find a "negation of the negation", we can say that a process will tend towards recurring cycles. For example, the tendency of capitalist accumulation to lead to a crisis of over-production, a first "negation", which will then destroy accumulated value and create the conditions for a new boom, a second "negation", leads to repeated booms and busts.
These inferences, from "positive feedback" to instability and from "negation of the negation" to cyclicity, are legitimate, but they are neither scientific laws nor alternatives to empirical study. In neither case does the conclusion give us any certainties; there may be counter-tendencies, or longer-term processes which erode the foundations of the system. But that is to be expected of a concept so general, and is acceptable, if we know what positive content it does have.
On the other hand, Ollman's definition also highlights a real weakness of the view of dialectics as a set of forms: a lack of obvious importance. The applicability of the negation of the negation so interpreted, to processes with three or more phases etc., is relatively narrow. If dialectics was a revolution in logic, in the basic tools of thought, then it would be obvious why it was worth studying. But if it is a mere collection of general patterns, what is the advantage of using or even speaking of a distinct dialectical method?
Consider the definition we have already cited of contradiction, as "the incompatible development of different elements within the same relation". Or Ollman's definition of the unity of opposites (which, perhaps wrongly, he distinguishes from contradiction): "the process by which a radical change in the conditions surrounding two or more elements... produces a striking alteration... in their relations".
There does not appear to be any intrinsic relationship between these concepts, considered as forms. If we find an instance of the negation of the negation, we cannot thereby deduce the existence of a contradiction in the same process, absent the "law" we have already rejected that change itself requires contradiction. And vice versa; if we find a contradiction in a system, that does not automatically mean that the system's equilibrium will fall apart of its own accord, let alone that such a negation will in turn be negated.
Moreover, in finding a pattern in some process which has a dialectical form, we are at the same time choosing a mode of appearance of that process to consider. The two meanings of "form" we discussed are bound together. Since the patterns we are looking for are abstract and apply across different domains, and so cannot be given any precise material criteria, they only appear when we describe - formulate - a process in a certain way. The material reality of any given situation can be stated without dialectical terminology, just as Earth's decreasing albedo can be described in detail without necessarily seeing it as a positive feedback in a larger process of global warming. We need a reason, if not permission, to use the concepts of dialectics.
Thus, dialectics cannot be simply a catalog of forms and remain valuable. In the absence of laws about all reality which dialectical forms express, we need something more. What's needed is a framework with which to unite these forms as a coherent object of study, to give them importance, and to assure us that they are more than mere forms of appearance, more than aesthetics.
The obvious starting point, as suggested in the previous post, is to say that dialectical patterns are characteristic of capitalism, and essential to understanding its functioning. That would fulfill all three of the requirements just listed. And unlike the idea that studies of nature and capitalism use distinct logical laws, this proposition does not require any strict separation of nature and humanity.
It does, however, leave open questions. The most fundamental is - characteristic why? What makes these forms essential to capitalism; why do we see them again and again? Upon the answer to this question depends the answer to a second question - why study the patterns, not simply the material specifics? What unique explanatory role do dialectical structures play?
 Science of Logic, p. 36.
 Algebra of Revolution, p. 274-5.
 George Novack, Logic of Marxism, p. 7.
 Dance of the Dialectic, p. 96.
 ibid., p. 17.
 ibid., p. 96.