  Given a hypersurface in a complex projective space, we prove that the
multidegrees of its toric polar map agree, up to sign, with the coefficients of
the Chern-Schwartz-MacPherson class of a distinguished open set, namely the
complement of the union of the hypersurface and the coordinate hyperplanes. In
particular, the degree of the toric polar map is given by the signed
topological Euler characteristic of the distinguished open set. For plane
curves, a precise formula for the degree of the toric polar map is obtained in
terms of local invariants. Finally, we construct families, in arbitrary
dimension, of irreducible hypersurfaces whose toric polar map is birational.
