  In recent years, the out-of-time-order correlator (OTOC) has emerged as a
diagnostic tool for information scrambling in quantum many-body systems. Here,
we present exact analytical results for the OTOC for a typical pair of random
local operators supported over two regions of a bipartition. Quite remarkably,
we show that this "bipartite OTOC" is equal to the operator entanglement of the
evolution and we determine its interplay with entangling power. Furthermore, we
compute long-time averages of the OTOC and reveal their connection with
eigenstate entanglement. For Hamiltonian systems, we uncover a hierarchy of
constraints over the structure of the spectrum and elucidate how this affects
the equilibration value of the OTOC. Finally, we provide operational
significance to this bipartite OTOC by unraveling intimate connections with
average entropy production and scrambling of information at the level of
quantum channels.
