I will discuss structural properties and entanglement of bipartite quantum states which are invariant under the action of the diagonal unitary group. This set of states unifies and generalizes several known classes of quantum states extensively studied in the literature. Their entanglement properties are related to a generalization of the notion of completely positive matrices. At the end, I will discuss the PPT squared conjecture, and show that these states satisfy this conjecture. This is joint work with Satvik Singh, see arXiv:2010.07898 and arXiv:2011.03809.