  Two-way relay is potentially an effective approach to spectrum sharing and
aggregation by allowing simultaneous bidirectional transmissions between
source-destinations pairs. This paper studies the two-way $2\times2\times2$
relay network, a class of four-unicast networks, where there are four
source/destination nodes and two relay nodes, with each source sending a
message to its destination. We show that without relay caching the total
degrees of freedom (DoF) is bounded from above by $8/3$, indicating that
bidirectional links do not double the DoF (It is known that the total DoF of
one-way $2\times2\times2$ relay network is $2$.). Further, we show that the DoF
of $8/3$ is achievable for the two-way $2\times2\times2$ relay network with
relay caching. Finally, even though the DoF of this network is no more than
$8/3$ for generic channel gains, DoF of $4$ can be achieved for a symmetric
configuration of channel gains.
