This work focuses on the abrupt changes that the application of large enough electric fields provokes in the internal structure of hematite/silicone oil suspensions. Experimental results reflect the existence of two well-defined structural patterns according to the strength of the field and the concentration of particles. At low electric fields, columns of particles between the electrodes can be observed when the concentration of solids exceeds a critical volume fraction. However, at higher fields, electrohydrodynamic convection and eventually electrophoretic migration take place, reflecting the relevance of the particle charge. A complete theoretical discussion is given to explain the origin of these so different behaviors. While the mismatch in the electrical properties (particularly, conductivity) of the solid and liquid phases, that is the Maxwell-Wagner polarization, can justify the chain-like structures of particles, it is necessary to take into an account the process of charge injection at the electrode/suspension interface to support the electrophoretic migration and deposition. The experimental conditions for which polarization or current effects predominate are elucidated in terms of the conductivity of the solid phase and the magnitude of the applied electric field