From such links with a self-crossing projection of a component, new infinite series of Borromean likns with twists are obtained. In a self-crossing point of the oriented component projection an n-twist is introduced, its orientation being used only for choosing the appropriate position of the twist. Note that the first series of Borromean links with twists could be also derived from Borromean rings by introducing identical twists in the crossing-points of two different components. Therefore, we could first get different infinite series of n-Borromean links without twists, and then introduce twists trying to preserve the Borromean property.