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Hypergraph neural networks (HyperGNNs) are a family of deep neural networks designed to perform inference on hypergraphs. HyperGNNs follow either a spectral or a spatial approach, in which a convolution or message-passing operation is conducted based on a hypergraph algebraic descriptor. While many HyperGNNs have been proposed and achieved state-of-the-art performance on broad applications, there have been limited attempts at exploring high-dimensional hypergraph descriptors (tensors) and joint node interactions carried by hyperedges. In this article, we depart from hypergraph matrix representations and present a new tensor-HyperGNN (T-HyperGNN) framework with cross-node interactions (CNIs). The T-HyperGNN framework consists of T-spectral convolution, T-spatial convolution, and T-message-passing HyperGNNs (T-MPHN). The T-spectral convolution HyperGNN is defined under the t-product algebra that closely connects to the spectral space. To improve computational efficiency for large hypergraphs, we localize the T-spectral convolution approach to formulate the T-spatial convolution and further devise a novel tensor-message-passing algorithm for practical implementation by studying a compressed adjacency tensor representation. Compared to the state-of-the-art approaches, our T-HyperGNNs preserve intrinsic high-order network structures without any hypergraph reduction and model the joint effects of nodes through a CNI layer. These advantages of our T-HyperGNNs are demonstrated in a wide range of real-world hypergraph datasets. The implementation code is available at https://github.com/wangfuli/T-HyperGNNs.git.