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A dual-polarized intelligent reflecting surface (IRS) can contribute to a better multiplexing of interfering wireless users. In this paper, we use this feature to improve the performance of dual-polarized massive multiple-input multiple-output (MIMO) with non-orthogonal multiple access (NOMA) under imperfect successive interference cancellation (SIC). By considering the downlink of a multi-cluster scenario, the IRSs assist the base station (BS) to multiplex subsets of users in the polarization domain. Our novel strategy alleviates the impact of imperfect SIC and enables users to exploit polarization diversity with near-zero inter-subset interference. To this end, the IRSs are optimized to mitigate transmissions originated at the BS from the interfering polarization. The formulated optimization is transformed into quadratic constrained quadratic sub-problems, which makes it possible to obtain the optimal solution via interior-points methods. We also derive analytically a closed-form expression for the users' ergodic rates by considering large numbers of reflecting elements. This is followed by representative simulation examples and comprehensive discussions. The results show that when the IRSs are large enough, the proposed scheme always outperforms conventional massive MIMO-NOMA and MIMO-OMA systems even if SIC error propagation is present. It is also confirmed that dual-polarized IRSs can make cross-polar transmissions beneficial to the users, allowing them to improve their performance through diversity.