As discussed in the comments, this stuff is more commonly described using the language of fans. A (convex, polyhedral) cone in $\mathbb{R}^{d}$ is the intersection of finitely-many half-spaces through the origin. A (polyhedral) fan is a collection of cones that intersect properly: the intersection of any two cones in the fan is again a cone in the fan which is a common face of both cones. A fan is complete if the union of all the cones in the fan is all of $\mathbb{R}^d$. By intersecting a complete fan with a sphere centered at the origin we obtain a spherical complex in the sense of the question-asker. This procedure should also be reversible by "coning over" a spherical complex. Cones have dual cones, ~~and in this way we get dual fans. This gives the desired combinatorial duality for spherical complexes.~~

**EDIT**: Whoops, now I am actually less sure of exactly how duality of cones leads to duality of fans. The dual cones of a fan will not fit together into a fan. So this does not answer the question. (Of course, if our fan happens to be polytopal, then we can use polar duality of polytopes.)

**EDIT 2**: I asked Vic Reiner about this question, and he gave me a lot of good information. He pointed out that the question of the existence of "dual" CW complexes is a difficult and subtle point, as discussed for instance in this other MO question. However, for a PL cell decomposition of $\mathbb{S}^d$ there exists a dual PL cell decomposition of $\mathbb{S}^d$ with a dual face lattice, as proved in Proposition 4.7.26(iv) on pg. 214 of the "Oriented Matroids" book by Björner et al. The spherical complexes you describe (a.k.a. polyhedral fans) will certain be PL, but this result still does not quite answer your question because it is not clear that the dual PL cell decomposition will correspond to a polyhedral fan.

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