Fermi-Löwdin orbitals (FLOs) are a special set of localized orbitals, which have become commonly used in combination with the Perdew-Zunger self-interaction correction (SIC) in the FLO-SIC method. The FLOs are obtained for a set of occupied orbitals by specifying a classical position for each electron. These positions are known as Fermi-orbital descriptors (FODs), and they have a clear relation to chemical bonding. In this study, we show how FLOs and FODs can be used to initialize, interpret, and justify SIC solutions in a common chemical picture, both within FLO-SIC and in traditional variational SIC, and to locate distinct local minima in either of these approaches. We demonstrate that FLOs based on Lewis theory lead to symmetry breaking for benzene-the electron density is found to break symmetry already at the symmetric molecular structure-while ones from Linnett's double-quartet theory reproduce symmetric electron densities and molecular geometries. Introducing a benchmark set of 16 planar cyclic molecules, we show that using Lewis theory as the starting point can lead to artifactual dipole moments of up to 1 D, while Linnett SIC dipole moments are in better agreement with experimental values. We suggest using the dipole moment as a diagnostic of symmetry breaking in SIC and monitoring it in all SIC calculations. We show that Linnett structures can often be seen as superpositions of Lewis structures and propose Linnett structures as a simple way to describe aromatic systems in SIC with reduced symmetry breaking. The role of hovering FODs is also briefly discussed.