The effect of the preference for nonparallel alignment without a preferred twist sense of neighboring headless spins is examined through computer simulations on extended Maier–Saupe models on the simple cubic lattice with/without next-nearest-neighbor interactions. Phase diagrams containing four phases are constructed for a varying degree of the preference for the local twist and for a strength of the next-nearest-neighbor interaction. It is shown that the preference for the local twist brings about not only the instability for the nematic (uniaxial) order but also that for a spatial order to reduce the frustration arising from the local twist. The next-nearest-neighbor interaction can produce even a weak spatial order of local chirality. The results are discussed in relation to the formation of a liquid crystalline gyroid phase and chiral phases in ensembles consisting of antispindle molecules that are axially symmetric and achiral.