While it is widely recognised that linear (structural) VARs may fail to capture important aspects of economic time series, the use of nonlinear SVARs has to date been almost entirely confined to the modelling of stationary time series, because of a lack of understanding as to how common stochastic trends may be accommodated within nonlinear models. This has unfortunately circumscribed the range of series to which such models can be applied -- and/or required that these series be first transformed to stationarity, a potential source of misspecification -- and prevented the use of long-run identifying restrictions in these models. To address these problems, we develop a flexible class of additively time-separable nonlinear SVARs, which subsume models with threshold-type endogenous regime switching, both of the piecewise linear and smooth transition varieties. We extend the Granger--Johansen representation theorem to this class of models, obtaining conditions that specialise exactly to the usual ones when the model is linear. We further show that, as a corollary, these models are capable of supporting the same kinds of long-run identifying restrictions as are available in linearly cointegrated SVARs.