The vortex-induced vibration of long cylinders in sheared flows is usually a mixture of standing and traveling waves. Understanding how the waveform is developed is of significant scientific and engineering interest. However, the interference between the traveling wave components makes the response patterns hard to interpret. Therefore, a spatial–temporal domain decomposition may offer a new perspective for the problem. This paper presents an eigenwave analysis method based on two-dimensional singular value decomposition, aiming to discover the hidden spatial–temporal modes. The application to experimental data shows its capability to separate the traveling and standing wave components, revealing the alternating dominance of these two waveforms along the span. It can also decompose the motion into two intrinsic traveling waves propagating in opposite directions, evolving and interacting in space and time. Based on the amplitude profiles of these waves, we locate the power-in/out regions and evaluate the lock-in range of reduced velocity as 4.73 to 7.68. It also reveals that the varying phase difference between multi-frequency modes, instead of the mode transition, is the main reason for the envelope fluctuation. The findings suggest that the proposed method is a useful data-driven technique extending the current toolkit for research on related topics.