  The origin of the $3 \times 1$ reconstruction observed in epitaxial
LaCoO$_{3}$ films on SrTiO$_3(001)$ is assessed by using first-principles
calculations including a Coulomb repulsion term. We compile a phase diagram as
a function of the oxygen pressure, which shows that ($3 \times 1$)-ordered
oxygen vacancies (LaCoO$_{2.67}$) are favored under commonly used growth
conditions, while stoichiometric films emerge under oxygen-rich conditions.
Growth of further reduced LaCoO$_{2.5}$ brownmillerite films is impeded by
phase separation. We report two competing ground-state candidates for
stoichiometric films: a semimetallic phase with $3 \times 1$
low-spin/intermediate-spin/intermediate-spin magnetic order and a
semiconducting phase with intermediate-spin magnetic order. This demonstrates
that tensile strain induces ferromagnetism even in the absence of oxygen
vacancies. Both phases exhibit an intriguing ($3 \times 1$)-reconstructed
octahedral rotation pattern and accordingly modulated La-La distances. In
particular, charge and bond disproportionation and concomitant orbital order of
the $t_{2g}$ hole emerge at the Co sites that are also observed for unstrained
bulk LaCoO$_3$ in the intermediate-spin state and explain structural data
obtained by x-ray diffraction at elevated temperature. Site disproportionation
drives a metal-to-semiconductor transition that reconciles the
intermediate-spin state with the experimentally observed low conductivity
during spin-state crossover without Jahn-Teller distortions.
