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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.