A result from particle tracking states that, after a microbunched electron beam is kicked, its trajectory changes while the orientation of the microbunching wavefront remains as before. Experiments at the LCLS showed that radiation in the kicked direction is produced practically without suppression. This could be explained if the orientation of the microbunching wavefront is readjusted along the kicked direction. In previous papers we showed that when the evolution of the electron beam modulation is treated according to relativistic kinematics, the orientation of the microbunching wavefront in the ultrarelativistic asymptotic is always perpendicular to the electron beam velocity. There we refrained from using advanced theoretical concepts to explain or analyze the wavefront rotation. For example, we only hinted to the relation of this phenomenon with the concept of Wigner rotation. This more abstract view of wavefront rotation underlines its elementary nature. The Wigner rotation is known as a fundamental effect in elementary particle physics. The composition of non collinear boosts does not result in a simple boost but, rather, in a Lorentz transformation involving a boost and a rotation, the Wigner rotation. Here we show that during the LCLS experiments, a Wigner rotation was actually directly recorded for the first time with a ultrarelativistic, macroscopic object: an ultrarelativistic electron bunch in an XFEL modulated at nm-scale of the size of about 10 microns. Here we point out the role of Wigner rotation in the analysis and interpretation of experiments with ultrarelativistic, microbunched electron beams in FELs. After the beam splitting experiment at the LCLS it became clear that, in the ultrarelativistic asymptotic, the projection of the microbunching wave vector onto the beam velocity is a Lorentz invariant, similar to the helicity in particle physics.