I originally posted this question on Stack Exchange, thinking it perhaps does not qualify as "research-level" but it received no answers... hopefully someone here can help.

The title pretty much sums up the question: what extra information do we need (or what is an example of sufficient information) on top of the face lattice in order to completely characterize a convex polytope? For example, would it be sufficient to provide a list of normal vectors associated with each facet for some embedding in $\mathbb{R}^d$?