Jockusch [J. Combin. Theory Ser. A 72 (1995), pp. 318–321] constructed an infinite family of centrally symmetric (cs, for short) triangulations of
3
3
-spheres that are cs-
2
2
-neighborly. Recently, Novik and Zheng [Adv. Math. 370 (2020), 16 pp.] extended Jockusch’s construction: for all
d
d
and
n
>
d
n>d
, they constructed a cs triangulation of a
d
d
-sphere with
2
n
2n
vertices,
Δ
n
d
\Delta ^d_n
, that is cs-
⌈
d
/
2
⌉
\lceil d/2\rceil
-neighborly. Here, several new cs constructions, related to
Δ
n
d
\Delta ^d_n
, are provided. It is shown that for all
k
>
2
k>2
and a sufficiently large
n
n
, there is another cs triangulation of a
(
2
k
−
1
)
(2k-1)
-sphere with
2
n
2n
vertices that is cs-
k
k
-neighborly, while for
k
=
2
k=2
there are
Ω
(
2
n
)
\Omega (2^n)
such pairwise non-isomorphic triangulations. It is also shown that for all
k
>
2
k>2
and a sufficiently large
n
n
, there are
Ω
(
2
n
)
\Omega (2^n)
pairwise non-isomorphic cs triangulations of a
(
2
k
−
1
)
(2k-1)
-sphere with
2
n
2n
vertices that are cs-
(
k
−
1
)
(k-1)
-neighborly. The constructions are based on studying facets of
Δ
n
d
\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
Δ
n
3
\Delta ^3_n
are shellable and an affirmative answer to Murai–Nevo’s question about
2
2
-stacked shellable balls is given.