Spectral clustering and co-clustering are well-known techniques in data analysis, and recent work has extended spectral clustering to square, symmetric tensors and hypermatrices derived from a network. We develop a new tensor spectral co-clustering method that applies to any non-negative tensor of data. The result of applying our method is a simultaneous clustering of the rows, columns, and slices of a three-mode tensor, and the idea generalizes to any number of modes. The algorithm we design works by recursively bisecting the tensor into two pieces. We also design a new measure to understand the role of each cluster in the tensor. Our new algorithm and pipeline are demonstrated in both synthetic and real-world problems. On synthetic problems with a planted higher-order cluster structure, our method is the only one that can reliably identify the planted structure in all cases. On tensors based on n-gram text data, we identify stop-words and semantically independent sets; on tensors from an airline-airport multimodal network, we find worldwide and regional co-clusters of airlines and airports; and on tensors from an email network, we identify daily-spam and focused-topic sets.