Motivated by Supervised Opinion Analysis, we propose a novel framework devoted to Structured Output Learning with Abstention (SOLA). The structure prediction model is able to abstain from predicting some labels in the structured output at a cost chosen by the user in a flexible way. For that purpose, we decompose the problem into the learning of a pair of predictors, one devoted to structured abstention and the other, to structured output prediction. To compare fully labeled training data with predictions potentially containing abstentions, we define a wide class of asymmetric abstention-aware losses. Learning is achieved by surrogate regression in an appropriate feature space while prediction with abstention is performed by solving a new pre-image problem. Thus, SOLA extends recent ideas about Structured Output Prediction via surrogate problems and calibration theory and enjoys statistical guarantees on the resulting excess risk. Instantiated on a hierarchical abstention-aware loss, SOLA is shown to be relevant for fine-grained opinion mining and gives state-of-the-art results on this task. Moreover, the abstention-aware representations can be used to competitively predict user-review ratings based on a sentence-level opinion predictor.