Summary: Motivated by the question of determining the difference in complexity between the graph-isomorphism and graph-automorphism problems, we study the relationship between the coset-intersection (C-INT) and group-intersection (G-INT) problems in permutation groups. We prove that G-INT is related to C-INT asymmetrically under Ogiwara and Watanabe's notions of left and right sets. This asymmetry suggests that, for given two permutation groups that intersect nontrivially, finding the least nonidentity element in the intersection is likely to be easier than finding the largest nonidentity element in the intersection under any lexicographic ordering. Also as a consequence, unless C-INT (resp., G-INT) $\in$P, C-INT (resp., G-INT) is not many-one reducible in polynomial time to any sparse set.