Self-organized, stable or dynamic, filamentary structures are known to form in dielectric barrier discharges (DBDs) operating in a glow regime. The common " classical " understanding of stable filamentary structures in DBDs is that a glow discharge filament forms at each half-cycle of the sinusoidal applied voltage at the same spatial location along the dielectric surface. The " memory " charges deposited by the previous filament on the dielectric surfaces are responsible for the ignition of a new discharge at the same location. On the other hand, the formation of a filamentary discharge at a given location inhibits discharge development in its vicinity because of charge spreading along the dielectric surface leading to a decrease of the gap voltage in this region. In this paper we show, using experiments and models that another self-organized filamentary regime is possible and coexists with the " classical " regime. In this regime, never reported before, discharge filaments at successive half cycles do not occur at the same location but are shifted by half a spatial period. We call this regime the " quincunx " regime. We also show that the mechanisms of this regime may be responsible for some dynamical aspects of self-organization in DBDs.