We carry out a detailed investigation of the statistical properties of the projected (angular) distribution of galaxy clusters obtained in cold dark matter (CDM) models with both Gaussian and skewed (i.e. non-Gaussian) primordial density fluctuations. We use large numerical simulations of these skewed CDM models to construct a set of simulations of the Lick catalogue. An objective cluster-finding algorithm is used to identify regions where the projected number density of galaxies in the catalogues exceeds some density threshold criterion. In this way we construct catalogues containing the angular positions and richnesses of real and simulated clusters which are suitable for statistical analysis. For Gaussian models, the overall number of clusters is too small in the standard CDM case compared to observations, but a model with higher normalization is in much better agreement; non-Gaussian models with negative initial skewness also fit the observed numbers fairly well. We compute the angular correlation function of clusters of different richness and find a strong dependence of the clustering amplitude on richness in all models. Even with a higher normalization, the Gaussian CDM model fails to produce sufficient large-scale cluster clustering. We also find that the Lick data are better reproduced only by a CDM model with negative initial skewness; initially skew-positive models fail to produce enough large-scale clustering. This conclusion is confirmed by two other statistical analyses; the properties of the minimal spanning tree and the multifractal scaling of the real clusters are much better reproduced by skew-negative CDM models. In particular, the small-scale self-similarity in the distribution of the richest real clusters turns out to be a crucial test, which is only passed by skew-negative models. We show that a skewness–variance relation of hierarchical type is followed by skew-negative models, as well as by the more evolved Gaussian model. The results for clusters suggest that the adoption of skewed primordial fluctuations is one way to enhance the production of large-scale structure in the CDM model. The skewed models seem to have some problems in reproducing the galaxy–galaxy two-point correlation function, but these problems may be connected with the limited spatial resolution of our simulations.