Chronological Order vs. Order of Exposition

Strauss's circular structure (going clockwise), in which the first (Heidegger) and last (Burke) chapter are thematically grouped, is connected by the hinge of the introduction, creating a Burke-Intro-Heidegger dyad on top mirrored by an Aristotle-Intro-Hobbes dyad on bottom.

Pairs in Natural Right and History

This is a schema of Natural Right and History. Heidegger-Burke-Rousseau-Plato are lined up under philosophy and Weber-Locke-Hobbes-Aristotle are under science. Each quadrant consists of a pair with an identical number of paragraphs - Heidegger-Burke (34 paragraphs), Weber-Locke (42 paragraphs), Rousseau-Plato (44 paragraphs), and Hobbes-Aristotle (40 paragraphs). Rousseau-Plato actually don't have the same number of paragraphs (44 and 48, respectively), but the last 4 paragraphs of Plato, which treat egalitarianism (in its ancient and modern types), belongs equally with either the Plato or Rousseau chapters (see 3.1, at bottom).

The way to think about the distinction between the "idea of philosophy" and the "idea of science" is that Hobbes-Aristotle are natural philosophers with positive doctrines, Weber-Locke reject natural philosophy but are methodological naturalists, Heidegger-Burke reject the possibility of philosophy (and science), and Rousseau-Plato are philosophers. Another way to look at it is that the top two quadrants reject intelligibility and the figures on the bottom assume some level of intelligibility. In that sense Rousseau and Hobbes are not typical moderns, hence the pairing with Plato and Aristotle, respectively.

For more discussion of pairs, see here. And for a discussion of intelligibility as the dividing line between ancient and modern, see here.