  BiFeO$_3$ is a technologically relevant multiferroic perovskite featuring
ferroelectricity and antiferromagnetism. Its lattice, magnetic, and
ferroelectric degrees of freedoms are coupled to its optically active
excitations and thus hold the potential to be reversible probed and controlled
by light. In this work, we combine ab initio density functional and many-body
perturbation theory methods with an extensive symmetry and atomic-orbital
analysis to describe and understand the electronic excited states spectrum and
its imprint on the optical absorption spectrum with quantitative accuracy and
qualitative insights. We find that the optical absorption spectrum of BiFeO$_3$
contain several strongly bound and spatially localized electronic transitions
in which the spin-degree of freedom is almost fully flipped. With our analysis
we thoroughly characterize these localized spin-flip transitions in terms of
the unusual crystal field splitting of Fe-$3d$ single-electron orbitals. Our
symmetry analysis further allows us to thoroughly explain how the spin content
and the energetic fine structure of these strongly bound excitons are dictated
by the interplay between crystal symmetry, electron-hole attraction, and the
spin-orbit coupling.
