This article deals with the stochastic modeling of industrial mistuned bladed disks. More specifically, a cost-efficient implementation of polynomial chaos expansion is proposed, it is dedicated to mathematical systems exhibiting permutation invariance or cyclic permutation invariance of their random variables. Significant gains are obtained in comparison to the classical implementation of polynomial chaos expansion since potentially costly evaluations of the investigated deterministic system are only required over a small subspace of the random space. The proposed methodology is detailed and validated on analytical test cases before it is applied to two mistuned bladed disks models. First, computations carried out with a simplified bladed disk model allow an in-depth comparison between Monte Carlo simulations, previously published results with a standard polynomial chaos expansion and the proposed methodology. The latter is then employed to assess the influence of mistuning on the eigenfrequencies and amplification magnification of an industrial compressor stage. It is evidenced that in comparison to the standard polynomial chaos expansion, the proposed methodology yields computational gains of the same order of magnitude as the ones obtained going from Monte Carlo simulations to polynomial chaos expansion. Alternately, the proposed methodology may be employed to significantly increase the accuracy of the standard polynomial chaos expansion while featuring an identical computational cost.