  In a $K$-pair-user two-way interference channel (TWIC), $2K$ messages and
$2K$ transmitters/receivers form a $K$-user IC in the forward direction ($K$
messages) and another $K$-user IC in the backward direction which operate in
full-duplex mode. All nodes may interact, or adapt inputs to past received
signals. We derive a new outer bound to demonstrate that the optimal degrees of
freedom (DoF, also known as the multiplexing gain) is $K$: full-duplex
operation doubles the DoF, but interaction does not further increase the DoF.
We next characterize the DoF of the $K$-pair-user TWIC with a MIMO, full-duplex
relay. If the relay is non-causal/instantaneous (at time $k$ forwards a
function of its received signals up to time $k$) and has $2K$ antennas, we
demonstrate a one-shot scheme where the relay mitigates all interference to
achieve the interference-free $2K$ DoF. In contrast, if the relay is causal (at
time $k$ forwards a function of its received signals up to time $k-1$), we show
that a full-duplex MIMO relay cannot increase the DoF of the $K$-pair-user TWIC
beyond $K$, as if no relay or interaction is present. We comment on reducing
the number of antennas at the instantaneous relay.
