The spin-orbit interaction was found to split the cyclotron resonance of heavy holes confined in high-mobility, compressively strained germanium quantum wells. The interference between coherent spin-split cyclotron resonances was tracked on picosecond time scales using terahertz time-domain spectroscopy. Analysis in the time domain, or using a time-frequency decomposition based on the Gabor-Morlet wavelet, was necessary when the difference between cyclotron frequencies was comparable to the linewidth. The cubic Rashba spin-orbit coefficient beta was determined via two methods: (i) the magnetic-field dependence of the cyclotron frequencies, and (ii) the spin-resolved subband densities. An enhanced beta and spin polarization was created by tailoring the strain to enhance the spin-orbit interaction. The amplitude modulation of the narrow, interfering cyclotron resonances is a signature of spin coherences persisting for more than 10 ps.