Background: Most people have strong left-brain lateralisation for language, with a minority showing right- or bilateral language representation. On some receptive language tasks, however, lateralisation appears to be reduced or absent. This contrasting pattern raises the question of whether and how language laterality may fractionate within individuals. Building on our prior work, we postulated (a) that there can be dissociations in lateralisation of different components of language, and (b) these would be more common in left-handers. A subsidiary hypothesis was that laterality indices will cluster according to two underlying factors corresponding to whether they involve generation of words or sentences, vs. receptive language. Methods: We tested these predictions in two stages: At Step 1 an online laterality battery (Dichotic listening, Rhyme Decision and Word Comprehension) was given to 621 individuals (56% left-handers); At Step 2, functional transcranial Doppler ultrasound (fTCD) was used with 230 of these individuals (51% left-handers). 108 left-handers and 101 right-handers gave usable data on a battery of three language generation and three receptive language tasks. Results: Neither the online nor fTCD measures supported the notion of a single language laterality factor. In general, for both online and fTCD measures, tests of language generation were left-lateralised. In contrast, the receptive tasks were at best weakly left-lateralised or, in the case of Word Comprehension, slightly right-lateralised. The online measures were only weakly correlated, if at all, with fTCD measures. Most of the fTCD measures had split-half reliabilities of at least .7, and showed a distinctive pattern of intercorrelation, supporting a modified two-factor model in which Phonological Decision (generation) and Sentence Decision (reception) loaded on both factors. The same factor structure fitted data from left- and right-handers, but mean scores on the two factors were lower (less left-lateralised) in left-handers. Conclusions: There are at least two factors influencing language lateralization in individuals, but they do not correspond neatly to language generation and comprehension. Future fMRI studies could help clarify how far they reflect activity in specific brain regions.