Sleep apnoea hypopnea syndrome ( SAHS) is a serious sleep disorder affecting a large percentage of the population. Apnoea/hypopnea and electroencephalographic-arousal ( EEGA) events occur frequently in SAHS patients. These events significantly disturb the sleep architecture, as revealed through nocturnal EEG signals. Even though EEG carries vital information on the state of the brain, its use in clinical SAHS diagnosis is limited mainly to routine sleep staging. In this paper, we address this issue. We propose a novel measure, called the inter-hemispheric asynchrony (psi (a -> b)), to capture EEG-symmetry changes associated with a transition a -> b between the brain states 'a' and 'b'. Our work takes into account macro-states such as the traditional sleep stages, and micro-states such as EEGA and apnoea/hypopnea events. We measured EEG data using electrodes C4-A2 and C3-A1 of the International 10/20 System from 18 subjects undergoing polysomnography ( PSG) testing. These electrode pairs are symmetrical about the brain mid-line and allow us to discern any hemispheric EEG asymmetry. EEG data were segmented and filtered into classical bands delta( 0.5-4 Hz), theta( 4.1-8 Hz), alpha( 8.1-12 Hz) and beta( 12.1-16 Hz). Then they were further categorized according to the particular sleep state of their origin. Spectral correlation coefficients were computed between the EEG data from the two hemispheres and averaged over the overnight EEG recording. This was done for each frequency band and state of interest, and then the measure psi(a -> b) was computed. Results from the 18 subjects showed that psi(a -> b) increased significantly ( p < 0.05) when the sleep state changed from NREM to REM, in all the frequency bands considered. Similarly, within both NREM and REM macro-states, psi(a -> b) changes significantly ( p < 0.1) with micro-state changes from the background state towards apnoea/ hypopnea and EEGA states. Extensive statistical analysis we conducted with the 18 subjects indicated that the measure psi(a -> b) provides a novel insight into the functional asymmetry of the brain during SAHS events.