It is shown that when relativistically intense ultrashort laser pulses are reflected from the boundary of a plasma with a near-critical density, the Doppler frequency shift leads to generation of intense radiation in both the high-frequency (up to the x-ray) and low-frequency (mid-infrared) ranges. The efficiency of energy conversion into the wavelength range above 3 µm can reach several percent, which makes it possible to obtain relativistically intense pulses in the mid-infrared range. These pulses are synchronized with high harmonics in the ultraviolet and x-ray ranges, which opens up opportunities for high-precision pump–probe measurements, in particular, laser-induced electron diffraction and transient absorption spectroscopy.