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Context: The Einasto model has become one of the most popular models for describing the density profile of dark matter haloes. There have been relatively few comprehensive studies on the dynamical structure of the Einasto model, mainly because only a limited number of properties can be calculated analytically. Aims: We want to systematically investigate the photometric and dynamical structure of the family of Einasto models over the entire model parameter space. Methods: We used the SpheCow code to explore the properties of the Einasto model. We systematically investigated how the most important properties change as a function of the Einasto index $n$. We considered both isotropic models and radially anisotropic models with an Osipkov-Merritt orbital structure. Results: We find that all Einasto models with $n<\tfrac12$ have a formal isotropic or Osipkov-Merritt distribution function that is negative in parts of phase space, and hence cannot be supported by such orbital structures. On the other hand, all models with larger values of $n$ can be supported by an isotropic orbital structure, or by an Osipkov-Merritt anisotropy, as long as the anisotropy radius is larger than a critical value. This critical anisotropy radius is a decreasing function of $n$, indicating that less centrally concentrated models allow for a larger degree of radial anisotropy. Conclusions: Studies of the structure and dynamics of models for galaxies and dark matter haloes should not be restricted to completely analytical models. Numerical codes such as SpheCow can help open up the range of models that are systematically investigated. This applies to the Einasto model discussed here, but also to other proposed models for dark matter haloes, including different extensions to the Einasto model.