Cylindrical soils undergo complicated bifurcation behaviors due to the loss of symmetry. As a first step to model its symmetry, the dihedral group symmetry of their cross section is exploited in Chaps. 9 and 11. To exploit symmetry breaking in the axial direction, this chapter deals with a larger group D∞h (≅ O (2 ) × ℤ2), which denotes the combination of upside-down symmetry and axisymmetry of a cylindrical domain. Recursive bifurcation and mode switching are highlighted as important behaviors. The perfect system is recovered with reference to imperfect behaviors of cylindrical soils using the procedure advanced in Chap. 6. Group-theoretic bifurcation theory presented in Chap. 8 and its application to the dihedral group in Chap. 9 are foundations of this chapter. An extension to a larger symmetry group O(2) ×O(2) is to be given in Chap. 16 to detect patterns with high spatial frequencies.