  We analyze bipartite matrices and linear maps between matrix algebras, which
are respectively, invariant and covariant, under the diagonal unitary and
orthogonal groups' actions. By presenting an expansive list of examples from
the literature, which includes notable entries like the Diagonal Symmetric
states and the Choi-type maps, we show that this class of matrices (and maps)
encompasses a wide variety of scenarios, thereby unifying their study. We
examine their linear algebraic structure and investigate different notions of
positivity through their convex conic manifestations. In particular, we
generalize the well-known cone of completely positive matrices to that of
triplewise completely positive matrices and connect it to the separability of
the relevant invariant states (or the entanglement breaking property of the
corresponding quantum channels). For linear maps, we provide explicit
characterizations of the stated covariance in terms of their Kraus,
Stinespring, and Choi representations, and systematically analyze the usual
properties of positivity, decomposability, complete positivity, and the like.
We also describe the invariant subspaces of these maps and use their structure
to provide necessary and sufficient conditions for separability of the
associated invariant bipartite states.
