A number of different dependence scenarios can arise in the theory of multivariate extremes, entailing careful selection of an appropriate class of models. In the simplest case of bivariate extremes, a dichotomy arises: pairs of variables are either asymptotically dependent or are asymptotically independent. Most available statistical models are suitable for either one case or the other, but not both. The consequence is a stage in the inference that is not accounted for, but which may have large impact upon the subsequent extrapolation. Previous modelling strategies that address this problem are either applicable only on restricted parts of the domain, or appeal to multiple limit theories. We present a unified representation for multivariate extremes that encompasses a wide variety of dependence scenarios, and is applicable when at least one variable is large. The representation motivates a parametric statistical model that is able to capture either dependence class, and model structure therein. We implement a simple version of this model, and show that it offers good estimation capability over a variety of dependence structures.