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Using two-dimensional numerical simulations, interaction of a counter-propagating probe laser pulse with a relativistic electron mirror formed from an overcritical or gas plasma layer by accelerating laser is investigated. When a non-adiabatic laser pulse of relativistic amplitude falls normally at a thin plasma layer, a simultaneous longitudinal (relative to the axis of the laser beam) displacement of all electrons can occur under the action of the longitudinal component of the Lorentz force. With a sufficiently large field amplitude, this force accelerates electrons to relativistic velocities. As a result, a relativistic electron mirror with a diameter of the order of several microns and with a thickness of several nanometers or less can be formed. It is shown that both coherent and incoherent parts in the scattered radiation can be formed and their characteristics are studied. For the coherent part of the radiation, the shape of the reflected pulse resembles that of the counter-propagating pulse. The reflected pulse frequency is proportional to the normalized energy of the electrons in the relativistic electron mirror. For the incoherent part, the scattered radiation frequency is proportional to the square of the normalized electrons energy and can be considerably larger than that for the coherent radiation. It is shown that for the dimensionless accelerating laser amplitude of the order of 20 (laser power of about 200 TW), the electrons with the energy of hundreds of MeV are produced and the energy of the scattered quanta can achieve several MeVs with very high brightness of the radiation.