Fullerene cages have n trivalent vertices, 12 pentagonal faces, and (n-20)/2 hexagonal faces. The smallest cage in which all of the pentagons are surrounded by hexagons and thus isolated from each other has 60 vertices and is shaped like a soccer ball. The protein clathrin self-assembles into fullerene cages of a variety of sizes and shapes, including smaller ones with adjacent pentagons as well as larger ones, but the variety is limited. To explain the range of clathrin architecture and how these fullerene cages self-assemble, we proposed a hypothesis, the "head-to-tail exclusion rule" (the "Rule"). Of the 5769 small clathrin cage isomers with n< or =60 vertices and adjacent pentagons, the Rule permits just 15, three identified in 1976 and 12 others. A "weak version" of the Rule permits another 99. Based on cryo-electron tomography, Cheng et al. reported six raw clathrin fullerene cages. One was among the three identified in 1976. Here, (1) we identify the remaining five. (2) Four are new and are among the 12 others permitted by the Rule. (3) One, also new, is among the 99 weak version cages. (4) Of particular note, none of the remaining 5565 excluded cages has been identified. These findings provide powerful experimental confirmation of the Rule and the principle on which it is based. (5) Surprisingly, the newly identified clathrin cages are among the least symmetric of those permitted. (6) By devising a method for counting assembly paths, (7) we show that asymmetric cages can be assembled by larger numbers of paths, thus providing a kinetic explanation for the prevalence of asymmetric cages. (8) Finally, we show that operation during cage growth of the Rule greatly increases the likelihood of producing a closed fullerene cage, specifically one of those permitted, but efficient assembly still appears to require internal remodeling.