This paper examines the postulate of local isotropy in stratified homogeneous turbulence from a theoretical point of view. The study is based on a priori analysis of the evolution equations governing single-point turbulence statistics that are formally consistent with the Navier--Stokes equations. The Boussinesq approximation has been utilized to account for the effect of buoyancy---a simplifying assumption which constitutes an excellent approximation for the case considered here. The study concludes that the hypothesis of local isotropy is formally inconsistent with the Navier--Stokes equations in homogeneous stratified turbulence. An estimate is provided that suggests that local isotropy may constitute only a physically justifiable approximation in the limit of a clear-cut separation between the time scales associated with the imposed buoyancy and the turbulent eddy turnover time scale. This is unlikely to happen in most flows, at least those not too far from equilibrium. The results also suggest that the dynamical dependence of the small-scale turbulence on large-scale anisotropies associated with imposed density stratification is significantly stronger than that caused by an imposed mean straining.