This chapter explores in a theoretical manner the potential correlations, previously suggested by Heine et al. (2015), between hemispheric specialization in neurolinguistics and a dual organization of discourse. The dual processing of language is represented here by the pragma-syntax, the level at which grammatical units manifest shared mental representations in discursive memory and generate inferences. While it is widely accepted that the left hemisphere of the brain is of utmost importance for morphosyntactic structuring of discourse, we will argue that the right hemisphere is crucial in processing operations at the pragma-syntactic level. Following a review of the questions surrounding hemispheric specialization, and of the language phenomena it can affect, we will present the pragma-syntactic model of human grammar. This paper will then suggest how it can effectively describe the phenomena whose comprehension has been previously reported in a number of studies as being dependent on the intact activity of the right brain, notably: irony, indirect speech acts and connectives. © 2020 John Benjamins Publishing Company.