We study four operations defined on pairs of Young tableaux. Algorithms for
the first three involve the familiar procedures of jeu de taquin, row
insertion, and column insertion, respectively. The fourth operation is new,
although specialised versions have appeared previously. Like the other
three operations, this new operation may be computed with a set of local
rules in a growth diagram, and it preserves Knuth equivalence class. Each
of these four operations gives rise to an a priori distinct theory of
dual equivalence. We show that these four theories coincide. The four
operations are linked via the involutive tableau operations of
complementation and conjugation in illuminating commutative diagrams.