  We propose a cross-order Laplacian renormalization group (X-LRG) scheme for
arbitrary higher-order networks. The renormalization group is a pillar of the
theory of scaling, scale-invariance, and universality in physics. An RG scheme
based on diffusion dynamics was recently introduced for complex networks with
dyadic interactions. Despite mounting evidence of the importance of polyadic
interactions, we still lack a general RG scheme for higher-order networks. Our
approach uses a diffusion process to group nodes or simplices, where
information can flow between nodes and between simplices (higher-order
interactions). This approach allows us (i) to probe higher-order structures,
defining scale-invariance at various orders, and (ii) to propose a
coarse-graining scheme. We demonstrate our approach on controlled synthetic
higher-order systems and then use it to detect the presence of order-specific
scale-invariant profiles of real-world complex systems from multiple domains.
