I might have over-stressed Honneth’s break with procedural liberalism in my last post, casting Honneth’s and procedural liberalism as contrasting theories. While it’s true they’re different, it’s wrong to imply that Honneth wants to part ways entirely with Rawls and company. Rather, Honneth sees his “reconstructive theory” of justice as a development from procedural liberalism.
Honneth claims his reconstructive theory is built upon and supplements procedural liberalism. In fact Honneth sees it as necessary but not sufficient to establish justice. For him, “this does not obviate distributional justice, bu it does demote it from the decisive principle of justice to a dependent variable in relations of recognition.” 
For a better understanding of how he accomplishes this, in another essay in the same book Honneth discusses three spheres of freedom in Hegel. The first is about individual desire and preference, the second about the morality necessary for the protection of individual desires and preference, and the third is about the concrete necessities for securing the common good.  Of course this third sphere isn’t separate and distinct from the other two, but a dialectical emergence. The first sphere is the expression of an individual’s preferences and desires. The second sphere is the negative freedom (to use Isaiah Berlin’s language) of how far this expression can go in my cooperative agreement to secure my possessions and goods. The third sphere is the positive freedom – the cooperation, goods, and space – to actualize freedom.
I think of these three as individual, cooperative, and mutual, and it’s this third piece of mutuality that is key. Mutuality is, in a sense, the next step in securing personal well-being. Here emerges, however, the understanding that the individual and her well-being is wrapped up in that of others. It’s here that our original dependence upon others – intrinsic to the theory of recognition – is realized most fully. Therefore it’s here in the third sphere that our social-relatedness is recaptured.
 Axel Honneth, The I in We: Studies in the Theory of Recognition, trans. Joseph Ganahl (Malden, MA: Polity Press, 2012) 46.
 Ibid., 29.