The controlled generation and identification of quantum correlations, usually encoded in either qubits or continuous degrees of freedom, builds the foundation of quantum information science. Recently, more sophisticated approaches, involving a combination of two distinct degrees of freedom have been proposed to improve on the traditional strategies. Hyperentanglement describes simultaneous entanglement in more than one distinct degree of freedom, whereas hybrid entanglement refers to entanglement shared between a discrete and a continuous degree of freedom. In this work we propose a scheme that allows to combine the two approaches, and to extend them to the strongest form of quantum correlations. Specifically, we show how two identical, initially separated particles can be manipulated to produce Bell nonlocality among their spins, among their momenta, as well as across their spins and momenta. We discuss possible experimental realizations with atomic and photonic systems. Introduction.-Sharing quantum correlations between distant parties is an indispensable condition for most tasks in quantum communication [1]. In the most common scenario, quantum information is encoded into a single, well-controlled degree of freedom (DOF), such as spin, polarization, or external degrees of freedom [2,3]. In some cases, however, establishing entanglement among several DOF can provide a decisive advantage [4][5][6][7][8][9][10][11][12]. For example, so-called hyperentanglement, i.e., entanglement in multiple DOF [4], can improve the capacity of dense coding in linear optics [7], or enhance the performance of quantum teleportation [6]. Similarly, architectures using hybrid entanglement, i.e., entanglement across discrete and continuous variables [5,13], have been suggested as a promising platform for quantum information, being able to overcome the limitations posed by the finite detection efficiencies of traditional approaches to quantum cryptography and computing [8,12].