We consider the problem of Collaborative Permutation Recovery, i.e. recovering multiple permutations over objects (e.g. preference rankings over different options) from limited pairwise comparisons. We tackle both the problem of how to recover multiple related permutations from limited observations, and the active learning problem of which pairwise comparison queries to ask so as to allow better recovery. There has been much work on recovering single permutations from pairwise comparisons, but we show that considering several related permutations jointly we can leverage their relatedness so as to reduce the number of comparisons needed compared to reconstructing each permutation separately. To do so, we take a collaborative filtering / matrix completion approach and use a trace-norm or max-norm regularized matrix learning model. Our approach can also be seen as a collaborative learning version of Jamieson and Nowak's recent work on constrained permutation recovery, where instead of basing the recovery on known features, we learn the best features de novo.