  Bismuth ferrite is one of the most widely studied multiferroic materials
because of its large ferroelectric polarisation coexisting with magnetic order
at room temperature. Using density functional theory (DFT), we identify several
previously unknown polar and non-polar structures within the low-energy phase
space of perovskite-structure bismuth ferrite, BiFeO$_3$. Of particular
interest is a series of non-centrosymmetric structures with polarisation along
one lattice vector, combined with anti-polar distortions, reminiscent of
ferroelectric domains, along a perpendicular direction. We discuss possible
routes to stabilising the new phases using biaxial heteroepitaxial strain or
interfacial electrostatic control in heterostructures.
