We study the {1234, 3412}-pattern-replacement equivalence relation on the set S_n of permutations of length n, which is conceptually similar to the Knuth relation. In particular, we enumerate and characterize the nontrivial equivalence classes, or equivalence classes with size greater than 1, in S_n for n ≥ 7 under the {1234, 3412}-equivalence. This proves a conjecture by Ma, who found three equivalence relations of interest in studying the number of nontrivial equivalence classes of S_n under pattern-replacement equivalence relations with patterns of length 4, enumerated the nontrivial classes under two of these relations, and left the aforementioned conjecture regarding enumeration under the third as an open problem.