Jockusch [J. Combin. Theory Ser. A 72 (1995), pp. 318-321] constructed an infinite family of centrally symmetric (cs, for short) triangulations of 3-spheres that are cs-2-neighborly. Recently, Novik and Zheng [Adv. Math. 370 (2020), 16 pp.] extended Jockusch's construction: for all d and n > d, they constructed a cs triangulation of a d-sphere with 2n vertices, Δ^d_n, that is cs-⌈d/2⌉-neighborly. Here, several new cs constructions, related to Δ^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 Ω(2ⁿ) such pairwise non-isomorphic triangulations. It is also shown that for all k > 2 and a sufficiently large n, there are Ω(2ⁿ) 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 Δ^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 Δ³_n are shellable and an affirmative answer to Murai-Nevo's question about 2-stacked shellable balls is given.