The rapidly rotating magnetospheres at Jupiter and Saturn contain a nearequatorial thin current sheet over most local times known as the magnetodisc, resembling a wrapped-up magnetotail. The Pioneer, Voyager, Ulysses, Galileo, Cassini and New Horizons spacecraft at Jupiter and Saturn have provided extensive datasets from which to observationally identify local time asymmetries in these magnetodiscs. Imaging in the infrared and ultraviolet from ground-and space-based instruments have also revealed the presence of local time asymmetries in the aurora which therefore must map to local time asymmetries in the magnetosphere. Asymmetries are found in (i) the configuration of the magnetic field and magnetospheric currents, where a thicker disc is found in the noon and dusk sectors; (ii) plasma flows where the plasma flow has local time-dependent radial components; (iii) a thicker plasma sheet in the dusk sector. Many of these features are also reproduced in global MHD simulations. Several models have been developed to interpret these various observations and typically fall into two groups: ones which invoke coupling with the solar wind (via reconnection or viscous processes) and ones which invoke internal rotational processes operating inside an asymmetrical external boundary. In this paper we review these observational in situ findings, review the models which seek to explain them, and highlight open questions and directions for future work.