<p>Hierarchical Risk Parity methods address some of the limitations of the classical mean-variance approach to portfolio selection by deriving a hierarchical structure. These methods are based on hierarchical clustering techniques and the recursive bisection of an ordered list of assets. When the number of assets is large, computational time becomes a limitation. This paper finds invariants of the allocation produced by simple asset permutations. We also study the size of the decision space to improve the understanding of the allocation algorithm. Building on these results, we propose a fast hierarchical risk parity portfolio selection method that reduces computational time while ensuring a similar performance.</p>