Spontaneous formation of spinning pairs of particles, or torsions, is studied in a single-layer complex plasma crystal by reducing the discharge power at constant neutral gas pressure. At higher gas pressures, torsions spontaneously form below a certain power threshold. Further reduction of the discharge power leads to the formation of multiple torsions. However, at lower gas pressures the torsion formation is preceded by mode-coupling instability (MCI). The crystal dynamics are studied with the help of the fluctuation spectra of crystal particles' in-plane velocities. Surprisingly, the spectra of the crystal with torsions and MCI are rather similar and contain hot spots at similar locations on the (k,ω) plane, despite very different appearances of the respective particle trajectories. The torsion rotation speed is close (slightly below) to the maximum frequency of the in-plane compressional mode. When multiple torsions form, their rotation speeds are distributed in a narrow range slightly below the maximum frequency.