We analyse the global (rigid) symmetries that are realised on the bosonic fields of the various supergravity actions obtained from eleven-dimensional supergravity by toroidal compactification followed by the dualisation of some subset of fields. In particular, we show how the global symmetries of the action can be affected by the choice of this subset. This phenomenon occurs even with the global symmetries of the equations of motion. A striking regularity is exhibited by the series of theories obtained respectively without any dualisation, with the dualisation of only the Ramond-Ramond fields of the type IIA theory, with full dualisation to lowest degree forms, and finally for certain inverse dualisations (increasing the degrees of some forms) to give the type IIB series. These theories may be called the GLA, D, E and GLB series, respectively. It turns out that the scalar Lagrangians of the E series are sigma models on the symmetric spaces K(E11-D)E11-D (where K(G) is the maximal compact subgroup of G) and the other three series lead to models on homogeneous spaces K(G)GS. These can be understood from the E series in terms of the deletion of positive roots associated with the dualised scalars, which implies a group contraction. We also propose a constrained Lagrangian version of the even-dimensional theories exhibiting the full duality symmetry and begin a systematic analysis of abelian subalgebras. 1998 Elsevier Science B.V.