We report on a phenomenon of plasmonic dichroism observed in magnetic materials with transverse magnetization under excitation of surface plasmon polariton waves. The effect originates from the interplay of the two magnetization-dependent contributions to the material absorption, both of which are enhanced under plasmon excitation. Plasmonic dichroism is similar to circular magnetic dichroism, which is at the base of all-optical helicity-dependent switching (AO-HDS) but observed for linearly polarized light, and the dichroism acts upon in-plane magnetized films, where AO-HDS does not take place. We show by electromagnetic modeling that laser pulses exciting counter-propagating plasmons can be used to write +M or -M states in a deterministic way independent of the initial magnetization state. The presented approach applies to various ferrimagnetic materials with in-plane magnetization, exhibiting the phenomenon of all-optical switching of a thermal nature and broadens the horizons of their applications in data storage devices.