The Madden-Julian Oscillation (MJO) is often identified in data through a combination of empirical orthogonal function (EOF) analysis and spectral filtering. However, these approaches can struggle to isolate dynamical modes of variability that evolve on different timescales but that all project onto a common set of EOFs used to capture the spatial variability of the MJO. In this paper, we use a data-driven approach that incorporates information about the underlying dynamics of the tropical atmosphere and ocean to identify the dynamical eigenmodes of the MJO. Our approach reveals that the dynamics of the MJO can be described by two robust dynamical eigenmodes that capture its fast and slow-propagating features. These eigenmodes project strongly onto indices such as the Real-time Multivariate MJO index (RMM), which is shown to represent not a single dynamical mode, but rather an aggregation of variance associated with both the fast and slow propagating MJO modes, as well as a set of eigenmodes representing ENSO. Our results demonstrate that these three dynamical modes of variability characterize the strongest MJO events as transitions from destructive interference at early lead times to constructive interference at longer lead times. At long enough lead times, it is shown how RMM use is complicated by the projection of both the MJO and ENSO onto RMM.