We present a unified deep learning framework for the recognition of user identity and the recognition of imagined actions, based on electroencephalography (EEG) signals, for application as a brain-computer interface. Our solution exploits a novel shifted subsampling preprocessing step as a form of data augmentation, and a matrix representation to encode the inherent local spatial relationships of multi-electrode EEG signals. The resulting image-like data is then fed to a convolutional neural network to process the local spatial dependencies, and eventually analyzed through a bidirectional long-short term memory module to focus on temporal relationships. Our solution is compared against several methods in the state of the art, showing comparable or superior performance on different tasks. Specifically, we achieve accuracy levels above 90% both for action and user classification tasks. In terms of user identification, we reach 0.39% equal error rate in the case of known users and gestures, and 6.16% in the more challenging case of unknown users and gestures. Preliminary experiments are also conducted in order to direct future works towards everyday applications relying on a reduced set of EEG electrodes.