We applied a large-amplitude oscillatory shear deformation (LAOS) to polystyrene-block-polyisoprene-block-polystyrene which had nearly randomly oriented grains of body-center-cubic lattice of spherical microdomains (BCC-sphere) before applying LAOS. Mechanical and structural responses of the BCC-sphere were simultaneously measured as a function strain phase phi and number of oscillatory strain cycles N by means of shear stress measurements and strain-phase-resolved dynamic small-angle X-ray scattering (DSAXS). The DSAXS revealed the following pieces of evidence: (1) LAOS first leads the BCC-sphere to attain two sets of the specific orientations as defined by A- and A'-sphere and by B- and B'-sphere, where both A- and A'-sphere and B- and B'-sphere are in the mirror symmetry with respect to {112} lattice plane which is parallel to the shear plane (the OXZ-plane with the OX and OZ-axis defined as the shear direction and vorticity or neutral axis, respectively) for A- and A'-sphere or to the OXY-plane with the OY-axis defined as the velocity gradient direction for B- and B'-sphere. The two sets commonly have their < 111 > axis along the OX-axis. (2) Moreover, in parallel to attaining the specific orientations, both A- and A'-sphere and B- and B'-sphere are directed to form twinned BCC-sphere with their {112} lattice planes as the twin plane. While the events (1) and (2) described above were found to occur in the early stage of strain cycles (N <= 4), in the late stage of strain cycles (N > 4) the twinned BCC-lattice itself was found to undergo the oscillatory shear deformation in response to the applied shear strain under the fixed orientation of the {112} lattice plane and the < 111 > axis as described above. Moreover, we found that the shear deformations of the A- and A'-sphere under those fixed orientations are strikingly unequal due to dominance of the entropy elasticity of coronar block chains over the energy elasticity associated with the strain imposed on the lattice spacings. We elucidated that stress variations with phi and N, including nonlinear response with stress hardening, are closely related to the structural responses with phi and N, respectively, as described above.