We study four operations defined on pairs of tableaux.
Algorithms for the first three involve the familiar procedures
of jeu de taquin, row insertion, and column insertion.
The fourth operation, hopscotch, 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 an illuminating commutative diagram.