In the analysis of data sets consisting of (X, Y)-pairs, a tacit assumption is that each pair corresponds to the same observational unit. If, however, such pairs are obtained via record linkage of two files, this assumption can be violated as a result of mismatch error rooting, for example, in the lack of reliable identifiers in the two files. Recently, there has been a surge of interest in this setting under the term "Shuffled Data" in which the underlying correct pairing of (X, Y)-pairs is represented via an unknown permutation. Explicit modeling of the permutation tends to be associated with overfitting, prompting the need for suitable methods of regularization. In this paper, we propose an exponential family prior on the permutation group for this purpose that can be used to integrate various structures such as sparse and local shuffling. This prior turns out to be conjugate for canonical shuffled data problems in which the likelihood conditional on a fixed permutation can be expressed as product over the corresponding (X,Y)-pairs. Inference can be based on the EM algorithm in which the E-step is approximated by sampling, e.g., via the Fisher-Yates algorithm. The M-step is shown to admit a reduction from n^2 to n terms if the likelihood of (X,Y)-pairs has exponential family form. Comparisons on synthetic and real data show that the proposed approach compares favorably to competing methods.