Based on the joint investigation in 287 healthy volunteers (150 left-Handers (LH)) of language task-induced asymmetries and intrinsic connectivity strength of the sentence-processing supramodal network, we show that individuals with atypical rightward language lateralization (N = 30, 25 LH) do not rely on an organization that simply mirrors that of typical leftward lateralized individuals. Actually, the resting-state organization in the atypicals showed that their sentence processing was underpinned by left and right networks both wired for language processing and highly interacting by strong interhemispheric intrinsic connectivity and larger corpus callosum volume. Such a loose hemispheric specialization for language permits the hosting of language in either the left and/or right hemisphere as assessed by a very high incidence of dissociations across various language task-induced asymmetries in this group.