Abstract Permutation equivalence and permutation congruence are special cases of matrix equivalence and similarity. This paper introduces a new invariant—the Hermite invariant—for testing permutation equivalence, along with a method for computing it and an assessment of its complexity. Under a restricted definition, the complexity of the invariant becomes polynomial in the dimensions of the input matrices. The sufficiency of the invariant is discussed, and experimental results are given. These results suggest that the Hermite invariant is particularly good at distinguishing nonpermutation equivalent matrices with constant row and column sums.