You can generate the Bernoullis through the combinatorics of permutahedra and graphical interpretations of surjections presented in OEIS-A133314 (cf. A049019, A019538, A008292) weighted by the reciprocal integers. (See also MOQ-61252.)
This is equivalent to determining the reciprocal of the exponential generating function
$$\frac{e^t-1}{t}=1+\frac{1}{2}t+ \frac{1}{3}\frac{t^2}{2!}+ \frac{1}{4}\frac{t^3}{3!}+\cdots\;\;.$$
Naturally, it's an involution, so you can go in the reverse direction from the Bernoullis to the reciprocal integers by the same surjections weighted by the Bernoulli numbers.
Alternatively, consider the e.g.f. $ \displaystyle \frac{2}{1+e^{2x}}$ for $ \displaystyle a(n-1)= \frac{2^n-4^n}{n} B_n$ , which without the zeros and signs are the tangent or zag numbers A000182 (a different normalization gives A002105). The reciprocal e.g.f. encodes $ \displaystyle b(n) = 1,2^0,2^1,2^2, ... $, which can be used as signed weights of surjection mappings to reconstitute the $a(n)$. These mappings are in turn encoded in the signed, refined face (partition) polynomials of the permutahedra of A133314.
But, with the o.g.f., using the normalized Bernoulli numbers, compositional inversion (A134264), rather than reciprocation enters the picture and, therefore, weighted noncrossing partitions and Dyck lattice paths (and myriad other related combinatoric structures).
These number arrays can be related to volumes of structures, as well as the Bernoullis (see Noam Elkies).
Also see these papers relating the Bernoullis to permutations, decomposition of hypercube volumes, and quantum algebras (related to A119467 and A119879):
Hodges and Sukumar, Sukumar and Hodges, Hetyei.
(Edit 10/7/2015)
What I find most interesting about Theo's question and follow-up answer are his attempts to connect the reciprocal integers and the Bernoulli numbers through topology/geometry since the two sequences are intimately paired through the formalism of Appell Sheffer sequences, two pairs with Lie and quantum algebras written all over them, and to the algebraic topology/geometry of exotic spheres through the Kervaire-Milnor formula related to the order of homotopy groups and multiplicative (elliptic) genera.