We explore two-way interference channels (ICs) where there are forward and backward ICs with four independent messages: two associated with the forward IC and the other two with respect to the backward IC. For a linear deterministic model of this channel, we develop inner and outer bounds on the capacity region. As a consequence, we demonstrate that interaction across forward and backward channels enables a more beneficial use of the channels, thereby yielding strict capacity improvements over non-interactive independent transmission. Moreover, our novel outer bound establishes the characterization of channel regimes in which interaction has no bearing on sum capacity.