We present a novel perspective on the clustering of dark matter in phase space by defining the particle phase space average density ($P^2SAD$) as a two-dimensional extension of the two-point correlation function averaged within a certain volume in phase space. This statistics is a sensitive measure of small scale (sub-)structure of dark matter haloes. By analysing the structure of $P^2SAD$ in Milky-Way-size haloes using the Aquarius simulations, we find it to be nearly universal at small scales, i.e. small separations in phase space, where substructures dominate. This remarkable universality occurs across time and in regions of substantially different ambient densities (by nearly four orders of magnitude), with typical variations in $P^2SAD$ of a factor of a few. The maximum variations occur in regions where substructures have been strongly disrupted. The universality is also preserved across haloes of similar mass but diverse mass accretion histories and subhalo distributions. The universality is also broken at large scales, where the smooth dark matter distribution in the halo dominates. Although at small scales the structure of $P^2SAD$ is roughly described by a subhalo model, we argue that the simulation data is better fitted by a family of superellipse contours. This functional shape is inspired by a model that extends the stable clustering hypothesis into phase space. In a companion paper, we refine this model and show its advantages as a method to obtain predictions for non-gravitational signatures of dark matter.
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