The convection-coupled tropical atmospheric motions are highly nonlinear and multiscaled, and play a major role in weather and climate predictability in both the tropics and mid-latitudes. In this work, nonlinear Laplacian spectral analysis (NLSA) is applied to extract spatiotemporal modes of variability in tropical dynamics from satellite observations. Blending qualitative analysis of dynamical systems, singular spectrum analysis (SSA), and spectral graph theory, NLSA has been shown to capture intermittency, rare events, and other nonlinear dynamical features not accessible through classical SSA. Applied to 1983-2006 satellite infrared brightness temperature data averaged over the global tropical belt, the method reveals a wealth of spatiotemporal patterns, most notably the 30-90-day Madden-Julian oscillation (MJO). Using the Tropical Ocean Global Atmosphere Coupled Ocean Atmosphere Response Experiment period as an example, representative modes associated with the MJO are reconstructed. The recovered modes augment Nakazawa's classical hierarchical structure of intraseasonal variability with intermediate modes between the fundamental MJO envelope and super cloud clusters.