Many empirical networks are intrinsically polyadic, with interactions occurring within groups of agents of arbitrary size. There are, however, few flexible null models that can support statistical inference for such polyadic networks. We define a class of null random hypergraphs that hold constant both the node degree and edge dimension sequences, generalizing the classical dyadic configuration model. We provide a Markov Chain Monte Carlo scheme for sampling from these models, and discuss connections and distinctions between our proposed models and previous approaches. We then illustrate these models through a triplet of applications. We start with two classical network topics -- triadic clustering and degree-assortativity. In each, we emphasize the importance of randomizing over hypergraph space rather than projected graph space, showing that this choice can dramatically alter statistical inference and study findings. We then define and study the edge intersection profile of a hypergraph as a measure of higher-order correlation between edges, and derive asymptotic approximations under the stub-labeled null. Our experiments emphasize the ability of explicit, statistically-grounded polyadic modeling to significantly enhance the toolbox of network data science. We close with suggestions for multiple avenues of future work.