The possibility for superdislocations in L12-ordered alloys to cross-slip from (111) to (111) planes has been considered. The stress to induce homogeneously this cross-slip is very high. Cross-slip can be locally activated, however, over an energy barrier of modest height, which is independent of the antiphase boundary energy but depends only on the anisotropy factorA.There are similarities to the kink-pair mechanism in the Peierls potential.