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Commonly used computational and analytical tools for fitness landscape analysis of optimization problems require identifying a distance metric that characterizes the similarity of different solutions to the problem. For example, fitness distance correlation is Pearson correlation between solution fitness and distance to the nearest optimal solution. In this paper, we survey the available distance metrics for permutations, and use principal component analysis to classify the metrics. The result is aligned with existing classifications of permutation problem types produced through less formal means, including the A-permutation, R-permutation, and P-permutation types, and has also identified subtypes. The classification can assist in identifying appropriate metrics based on optimization problem feature for use in fitness landscape analysis. Implementations of all of the permutation metrics, and the code for our analysis, are available as open source.