<p>The renormalization group is a pillar of the theory of scaling, scale invariance and universality in physics. Recently, this tool has been adapted to complex networks with pairwise interactions through a scheme based on diffusion dynamics. However, as the importance of polyadic interactions in complex systems becomes more evident, there is a pressing need to extend the renormalization group methods to higher-order networks. Here we fill this gap and propose a Laplacian renormalization group scheme for arbitrary higher-order networks. At the heart of our approach is the introduction of cross-order Laplacians, which generalize existing higher-order Laplacians by allowing the description of diffusion processes that can happen on hyperedges of any order via hyperedges of any other order. This approach enables us to probe higher-order structures, define scale invariance at various orders and propose a coarse-graining scheme. We validate 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.</p>