Formation of the {100}<001> cube texture component has been reported extensively in many heavily deformed face-centered cubic (FCC) metals and alloys after annealing. The present study shows how to avoid the formation of strong cube texture on the surface of high-temperature annealed Ni foils subjected to large rolling reductions and also retain the high intensity of twin boundaries. The evolution of the surface texture of rolled and annealed foils are discussed in comparison to conventional rolling, shear, and recrystallization texture components. The stability of the cube and ND-rotated cube components in rolled foils are elaborated. In spite of discontinuous recrystallization mechanisms, a high fraction of rolling texture components is retained after annealing. Suppression of the cube component after annealing has been attributed to the constricted-annealing process and possible activation of low-energy twin boundaries. Annealing twins were found to form during the recrystallization process via a growth fault mechanism.