Evolution of the net magnetization M of a nanocomposite magnetic bilayer consisting of exchange-coupled soft ferromagnetic/antiferromagnetic films under a rotating magnetic field H has been investigated using an advanced magnetooptic-indicator-film technique. The magnetization reversal of the ferromagnetic film was found to proceed via an exchange-spring formation due to twisting of the layers of the spin system. Two magnetization-reversal modes were found. At high fields, M rotates synchronously with the field. At low fields, there are stages of M and H rotation in either the same or opposite directions. The revealed features of the magnetization-reversal processes are compared with those for soft/hard ferromagnetic bilayers. The experimental data are discussed in terms of the hybrid spin-spring nucleation and evolution with allowance for the unidirectional anisotropy affected by interface imperfections and crystal-lattice defects.