Is there a "nice" way to compute the signature of a smooth toric manifold of even complex dimension in terms of the moment polytope? By signature I mean in the sense of topology (see https://en.wikipedia.org/wiki/Signature_(topology)).

If one goes through all the machinery it is clear that the signature is encoded. I am asking if there is a some way you can "see" this quantity from the moment polytope, quickly and effortlessly. For example the topological Euler characteristic is the number of vertices.  The signature won't be as nice as this, certainly, but that is the spirit of what I am looking for (i.e. the less machinery the better). In complex dimension 2 the signature is also a function of the number of vertices by the minimal model program, but that seems to be a stroke of fortune owing to the low dimension.