Magnetization reversal in exchange-spring magnet films has been investigated by a first-order reversal curve (FORC) technique and vector magnetometry. In Fe/epitaxial-SmCo films, the reversal proceeds by a reversible rotation of the Fe soft layer, followed by an irreversible switching of the SmCo hard layer. The switching fields are clearly manifested by separate steps in both longitudinal and transverse hysteresis loops, as well as sharp boundaries in the FORC distribution. In FeNi/polycrystalline-FePt films, particularly with thin FeNi, the switching fields are masked by the smooth and step-free major loop. However, the FORC diagram still displays a distinct onset of irreversible switching and transverse hysteresis loops exhibit a pair of peaks, whose amplitude is larger than the maximum possible contribution from the FeNi layer alone. This suggests that the FeNi and FePt layers reverse in a continuous process via a vertical spiral. The successive versus continuous rotation of the soft/hard layer system is primarily due to the different crystal structure of the hard layer, which results in different anisotropies.