In The Savage Mind (1962), Lévi-Strauss makes his most politically charged philosophical argument. He attacks directly the assumption, common in Enlightenment and colonial thought, that there is a developmental scale running from primitive to civilized thought, and that Western scientific rationality sits at the top.
His alternative: there are two distinct modes of scientific thinking, both of which are genuinely scientific in the sense of being systematic, rigorous, and productive of real knowledge. He calls them the engineer and the bricoleur.
The engineer thinks in concepts abstracted from any particular context: general principles, universal laws, mathematical structures. Engineering builds from the top down: you identify the abstract requirements for a structure and then select or create the materials to meet them. Modern science is largely engineering in this sense.
The bricoleur (the tinker, the handyman) thinks with whatever is to hand: the specific objects, materials, and techniques that have accumulated from previous projects, each of which retains its particular character and history. Bricolage builds from the bottom up: you see what your available materials suggest and construct something that works with what you have. The result is not a pure realization of an abstract design but a complex, layered artifact whose structure is partly determined by the properties of its materials.
Lévi-Strauss argues that mythical thought is a form of bricolage. It uses the cultural materials available, including specific animals, specific field has, specific social arrangements, and weaves them into structural systems that organize experience and resolve contradictions. It is not less rigorous or less logical than scientific thought. It is differently oriented: toward the concrete, the particular, and the immediately available rather than toward the abstract and the general.
The political implication is important. If mythical thought is a genuine mode of scientific thinking rather than a primitive precursor to real science, then the Enlightenment narrative of progress from savage to civilized thought is wrong. The Amazonian shaman analyzing the properties of hundreds of medicinal plants and organizing them into a complex taxonomic system is doing the same intellectual work as the Western botanist, just using different categories and a different relationship to the concrete. The categories are different but the cognitive capacity is identical.
Lévi-Strauss is also famous for his structural analysis of kinship systems, which he developed in The Elementary Structures of Kinship (1949). He argued that kinship rules in all human societies are organized around the prohibition of incest and the exchange of women. He analyzed as a form of communication: women circulate between groups as signs in a system of social alliance, just as words circulate in a system of communication. This analysis was enormously influential and almost equally controversial, particularly its treatment of women as signs exchanged by men rather than as agents in their own right. Gayle Rubin's feminist critique of this in "The Traffic in Women" (1975) is one of the foundational texts of feminist theory.