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We have made a systematic theoretical survey of the competition between ferroelastic and ferroelectric instabilities in the family of halide-based perovskites of formula ABX3, where A is an alkali-metal ion, B is a Be, Mg, or Ca ion, and X is a halide ion. Initially we surveyed the whole series of such compounds, making a theoretical lattice-dynamical study using first-principles interionic potentials composed of a long-range pure Coulomb interaction between the spherically symmetric free ions, and a short-range component calculated by the Gordon-Kim approach from the overlapping free-ion charge densities. We then proceeded to examine in more detail three compounds, NaCaBr3, NaCaCl3, and NaCaF3, which manifested both ferroelectric (zone-center) and ferroelastic (zone-boundary) instabilities (there were no structures which showed zone-center instabilities alone). For these three systems we then proceeded to a full energy minimization. This was done by allowing all 40 ions in the lowest symmetry phase to relax independently. It was found that the most stable structure of all three compounds consisted of triply canted halide octahedra, turned through equal angles about all three Cartesian axes. In this phase all eigenfrequencies are real, implying absolute stability, and the ferroelectric instability has been removed. We also discuss the possibility of a ferroelectric, or near-ferroelectric, intermediate phase.