  Hexagonal BaTiO_3 undergoes a structural phase transition to an orthorhombic
C222_1 phase at T_0 = 222 K. The transition is driven by a soft optical mode
with E_2u symmetry whose couplings force the appearance of a spontaneous E_2g
strain (improper ferroelastic character). Staying within the same E_2u
subspace, the system could in principle settle into a second (not observed)
orthorhombic phase (Cmc2_1). We have carried out a first-principles
investigation of these questions, studying the structure of the existing C222_1
and the "virtual" Cmc2_1 phases, and describing the spontaneous E_2g strain in
accord with the experimental observations. In addition, we show that the
occurrence of C222_1 instead of Cmc2_1 cannot be explained by the E_2u soft
modes themselves and, therefore, must be related to their couplings with
secondary order parameters. A more detailed analysis proves that the E_2g
strains do not account for the experimental preference.
