  In 1995, Josckusch constructed an infinite family of centrally symmetric (cs,
for short) triangulations of $3$-spheres that are cs-$2$-neighborly. Recently,
Novik and Zheng extended Jockusch's construction: for all $d$ and $n>d$, they
constructed a cs triangulation of a $d$-sphere with $2n$ vertices,
$\Delta^d_n$, that is cs-$\lceil d/2\rceil$-neighborly. Here, several new cs
constructions, related to $\Delta^d_n$, are provided. It is shown that for all
$k>2$ and a sufficiently large $n$, there is another cs triangulation of a
$(2k-1)$-sphere with $2n$ vertices that is cs-$k$-neighborly, while for $k=2$
there are $\Omega(2^n)$ such pairwise non-isomorphic triangulations. It is also
shown that for all $k>2$ and a sufficiently large $n$, there are $\Omega(2^n)$
pairwise non-isomorphic cs triangulations of a $(2k-1)$-sphere with $2n$
vertices that are cs-$(k-1)$-neighborly. The constructions are based on
studying facets of $\Delta^d_n$, and, in particular, on some necessary and some
sufficient conditions similar in spirit to Gale's evenness condition. Along the
way, it is proved that Jockusch's spheres $\Delta^3_n$ are shellable and an
affirmative answer to Murai--Nevo's question about $2$-stacked shellable balls
is given.
