In the inverted-band regime of HgTe/CdTe quantum wells, the lowest Landau level of the lowest conduction subband and the highest Landau level of the topmost valence subband are predicted to cross at a critical magnetic field B. We study this crossing experimentally with far-infrared Fourier-transform spectroscopy in a gated HgTe/CdTe quantum well with tunable electron density. The crossing point is identified by a characteristic exchange of oscillator strength between the two transitions involved, one being a cyclotron resonance, the other an intersubband resonance. The experimental resonance positions, the oscillator strengths as well as the value of B, are in good agreement with theoretical results of a 6 x 6 { k·p} model evaluated for the [211] growth direction.