  Zarankiewicz's Conjecture (ZC) states that the crossing number cr$(K_{m,n})$
equals $Z(m,n):=\floor{\frac{m}{2}} \floor{\frac{m-1}{2}} \floor{\frac{n}{2}}
\floor{\frac{n-1}{2}}$. Since Kleitman's verification of ZC for $K_{5,n}$ (from
which ZC for $K_{6,n}$ easily follows), very little progress has been made
around ZC; the most notable exceptions involve computer-aided results. With the
aim of gaining a more profound understanding of this notoriously difficult
conjecture, we investigate the optimal (that is, crossing-minimal) drawings of
$K_{5,n}$. The widely known natural drawings of $K_{m,n}$ (the so-called
Zarankiewicz drawings) with $Z(m,n)$ crossings contain antipodal vertices, that
is, pairs of degree-$m$ vertices such that their induced drawing of $K_{m,2}$
has no crossings. Antipodal vertices also play a major role in Kleitman's
inductive proof that cr$(K_{5,n}) = Z(5,n)$. We explore in depth the role of
antipodal vertices in optimal drawings of $K_{5,n}$, for $n$ even. We prove
that if {$n \equiv 2$ (mod 4)}, then every optimal drawing of $K_{5,n}$ has
antipodal vertices. We also exhibit a two-parameter family of optimal drawings
$D_{r,s}$ of $K_{5,4(r+s)}$ (for $r,s\ge 0$), with no antipodal vertices, and
show that if $n\equiv 0$ (mod 4), then every optimal drawing of $K_{5,n}$
without antipodal vertices is (vertex rotation) isomorphic to $D_{r,s}$ for
some integers $r,s$. As a corollary, we show that if $n$ is even, then every
optimal drawing of $K_{5,n}$ is the superimposition of Zarankiewicz drawings
with a drawing isomorphic to $D_{r,s}$ for some nonnegative integers $r,s$.
