We designed a nanocomposite consisting of CdSe nanoplatelets dispersed in the form of short stacks in the polybutadiene domains of a polystyrene-polybutadiene-polystyrene (SBS) thermoplastic elastomer matrix. Under strain, the material displays reversible, macroscopic anisotropic properties, e.g. the fluorescence signal. We present here a structural study of the composite under stretching, by in situ high-resolution X-ray scattering using synchrotron radiation. Modelling the scattering signal allows us to monitor the evolution of both the matrix and the platelets under strain. In particular, we show that the strain "reshuffles" the platelet stacks, which tilt their long axis from parallel to the plane of the microstructure lamellae at rest to perpendicular to this plane at high strain, at the same time breaking into smaller pieces, more easily accommodated in the soft butadiene domains. This reshuffling is fully reversed after strain relaxation. Moreover, it can be prevented by adding free oleic acid, which reinforces the interactions between the platelets in the stacks.