Electroencephalografic (EEG) data are complex multi-dimensional time-series that are very useful in many applications, from diagnostics to driving brain-computer interface systems. Their classification is still a challenging task, due to the inherent within- and between-subject variability and their low signal-to-noise ratio. On the other hand, the reconstruction of raw EEG data is even more difficult because of the high temporal resolution of these signals. Recent literature has proposed numerous machine and deep learning models that could classify, e.g., different types of movements, with an accuracy in the range 70% to 80% (with 4 classes). On the other hand, a limited number of works targeted the reconstruction problem, with very limited results. In this work, we propose vEEGNet, a DL architecture with two modules, i.e., an unsupervised module based on variational autoencoders to extract a latent representation of the data, and a supervised module based on a feed-forward neural network to classify different movements. To build the encoder and the decoder of VAE we exploited the well-known EEGNet network. We implemented two slightly different architectures of vEEGNet, thus showing state-of-the-art classification performance, and the ability to reconstruct both low-frequency and middle-range components of the raw EEG. Although preliminary, this work is promising as we found out that the low-frequency reconstructed signals are consistent with the so-called motor-related cortical potentials, well-known motor-related EEG patterns and we could improve over previous literature by reconstructing faster EEG components, too. Further investigations are needed to explore the potentialities of vEEGNet in reconstructing the full EEG data, generating new samples, and studying the relationship between classification and reconstruction performance.