This paper aims to recover a multi-subspace matrix from permuted data: given a matrix, in which the columns are drawn from a union of low-dimensional subspaces and some columns are corrupted by permutations on their entries, recover the original matrix. The task has numerous practical applications such as data cleaning, integration, and de-anonymization, but it remains challenging and cannot be well addressed by existing techniques such as robust principal component analysis because of the presence of multiple subspaces and the permutations on the elements of vectors. To solve the challenge, we develop a novel four-stage algorithm pipeline including outlier identification, subspace reconstruction, outlier classification, and unsupervised sensing for permuted vector recovery. Particularly, we provide theoretical guarantees for the outlier classification step, ensuring reliable multi-subspace matrix recovery. Our pipeline is compared with state-of-the-art competitors on multiple benchmarks and shows superior performance.