  It was recently shown \cite{opposite} that systems with opposite
thermodynamic arrows of time could have moderate mutual interaction with
neither destroying the order of the other. Such interaction includes signaling.
Signals, however, may give rise to causal paradoxes, suggesting that "moderate"
interaction may be impossible. Using the two-time boundary condition framework,
we resolve the paradoxes. In one example, at a macroscopic level, we establish
the existence of solutions to the appropriate boundary value problem. This
result is extended to a class of microscopic problems. We also produce an
example in which microscopic data are given and there is no solution. This is a
different kind of resolution: there is no paradox because the events do not
happen. Finally we discuss the differences between these cases.
