A square-free monomial ideal $I$ of $k[x_1,\ldots,x_n]$ is said to be an
$f$-ideal if the facet complex and non-face complex associated with $I$ have
the same $f$-vector. We show that $I$ is an $f$-ideal if and only if its Newton
complementary dual $\widehat{I}$ is also an $f$-ideal. Because of this duality,
previous results about some classes of $f$-ideals can be extended to a much
larger class of $f$-ideals. An interesting by-product of our work is an
alternative formulation of the Kruskal-Katona theorem for $f$-vectors of
simplicial complexes.