A finite abstract polytope of rank 3 (an abstract polyhedron) consists of adjacency data for a collection of polygonal faces and their shared edges and vertices. This data is sufficient to uniquely determine a topological surface obtained by gluing polygonal disks together along their boundaries. If the resulting surface is homeomorphic to the sphere $S^2$, we say that the polyhedron is spherical.
I think we can define sphericalness for higher ranks inductively: If a rank-$n$ polytope $P$ has spherical facets (of rank $n-1$), then it determines (I think, up to topological equivalence) a gluing of $(n-1)$-dimensional balls to each other along their $S^{n-2}$ boundaries (in the manner of a CW complex). We say $P$ is spherical if the result is $S^{n-1}$. (A necessary condition for this is that $P$'s abstract vertex figures are also spherical, so in fact all proper sections of $P$ are spherical. I think any abstract polytope that is "locally spherical" in this sense obtains a manifold structure.)
Thanks to the classification of closed surfaces, we can algorithmically check whether an abstract polyhedron is spherical just by checking whether it's orientable and computing its Euler characteristic. But what about higher ranks? Is there an algorithm that determines whether any given finite abstract polytope is spherical?