Hypergraphs are a powerful abstraction for modeling high-order interactions between a set of entities of interest and have been attracting a growing interest in the graph-learning literature. In particular, directed hypegraphs are crucial in their capability of representing real-world phenomena involving group relations where two sets of elements affect one another in an asymmetric way. Despite such a vast potential, an established, principled solution to tackle graph-learning tasks on directed hypergraphs is still lacking. For this reason, in this paper we introduce the Generalized Directed Hypergraph Neural Network (GeDi-HNN), the first spectral-based Hypergraph Neural Network (HNN) capable of seamlessly handling hypergraphs featuring both directed and undirected hyperedges. GeDi-HNN relies on a graph-convolution operator which is built on top of a novel complex-valued Hermitian matrix which we introduce in this paper: the Generalized Directed Laplacian⃗LN. We prove that ⃗LN generalizes many previously-proposed Laplacian matrices to directed hypergraphs while enjoying several desirable spectral properties. Extensive computational experiments against state-of-the-art methods on real-world and synthetically-generated datasets demonstrate the efficacy of our proposed HNN. Thanks to effectively leveraging the directional information contained in these datasets, GeDi-HNN achieves a relative-percentage-difference improvement of 7% on average (with a maximum improvement of 23.19%) on the real-world datasets and of 65.3% on average on the synthetic ones.