  Gr\"unbaum, Barnette, and Reay in 1974 completed the characterization of the
pairs $(f_i,f_j)$ of face numbers of $4$-dimensional polytopes.
  Here we obtain a complete characterization of the pairs of flag numbers
$(f_0,f_{03})$ for $4$-polytopes. Furthermore, we describe the pairs of face
numbers $(f_0,f_{d-1})$ for $d$-polytopes; this description is complete for
even $d\ge6$ except for finitely many exceptional pairs that are "small" in a
well-defined sense, while for odd $d$ we show that there are also "large"
exceptional pairs.
  Our proofs rely on the insight that "small" pairs need to be defined and to
be treated separately; in the $4$-dimensional case, these may be characterized
with the help of the characterizations of the $4$-polytopes with at most $8$
vertices by Altshuler and Steinberg (1984).
