Hemispheric lateralisation is a fundamental principle of functional brain organisation. We studied two core cognitive functions--language and visuospatial attention--that typically lateralise in opposite cerebral hemispheres. In this work, we tested both left- and right-handed participants on lexical decision-making as well as on symmetry detection by means of a visual half-field paradigm with various target--distractor combinations simultaneously presented to opposite visual fields. Laterality indexes were analysed using a behavioural metrics in single individuals as well as between individuals. We observed that lateralisation of language and visuospatial attention as well as their relationship generally followed a left-right profile, albeit with differences as a function of handedness and target-distractor combination. In particular, right-handed individuals tended towards a typical pattern whereas left-handed individuals demonstrated increased individual variation and atypical organisation. That the atypical variants varied as a function of target--distractor combination and thus interhemispheric communication underlines its dynamic role in characterising lateralisation properties. The data further revealed distinctive relationships between right-handedness and left-hemispheric dominance for language together with right-hemispheric dominance for visuospatial processing. Overall, these findings illustrate the role of broader mechanisms in supporting hemispheric lateralisation of cognition and behaviour, relying on common principles but controlled by internal and external factors.