Symmetric products of absolute Lipschitz retracts

We prove that the Qth symmetric product of an absolute $\lambda$-Lipschitz retract, equipped with the bottleneck matching metric, is an absolute $(3Q-2)\lambda$-Lipschitz retract. This answers a question of De Pauw about the target spaces of Almgren’s Q-valued maps. The main tool is a reconstruction theorem for bounded products of metric stars with finitely many rays, modulo a finite permutation group: these quotients are absolute 3-Lipschitz retracts, independently of the group and the number of coordinates.