  Perovskite bilayers with (111)-orientation combine a honeycomb lattice as a
key feature with the strongly correlated, multiorbital nature of electrons in
transition metal oxides. In a systematic DFT+$U$ study of (111)-oriented
(La$X$O$_3$)$_2$/(LaAlO$_3$)$_4$ superlattices, we establish trends in the
evolution of ground states versus band filling in (111)-oriented
(La$X$O$_3$)$_2$/(LaAlO$_3$)$_4$ superlattices, with $X$ spanning the entire
$3d$ transition metal series. The competition between local quasi-cubic and
global triangular symmetry triggers unanticipated broken symmetry phases, with
mechanisms ranging from Jahn-Teller distortions, to charge-, spin-, and
orbital-ordering. LaMnO$_3$, where spin-orbit coupling opens a sizable gap in
the Dirac-point Fermi surface, emerges as a topological Chern insulator.
