The expected possession value (EPV) of a soccer possession represents the likelihood of a team scoring or conceding the next goal at any time instance. In this work, we develop a comprehensive analysis framework for the EPV, providing soccer practitioners with the ability to evaluate the impact of observed and potential actions, both visually and analytically. The EPV expression is decomposed into a series of subcomponents that model the influence of passes, ball drives and shot actions on the expected outcome of a possession. We show we can learn from spatiotemporal tracking data and obtain calibrated models for all the components of the EPV. For the components related with passes, we produce visually-interpretable probability surfaces from a series of deep neural network architectures built on top of flexible representations of game states. Additionally, we present a series of novel practical applications providing coaches with an enriched interpretation of specific game situations. This is, to our knowledge, the first EPV approach in soccer that uses this decomposition and incorporates the dynamics of the 22 players and the ball through tracking data.