We prove that the Cartesian product of two directed cycles of lengths \(n_1\) and \(n_2\) contains an antidirected Hamilton cycle, and hence is decomposable into antidirected Hamilton cycles, if and only if \(\gcd(n_1, n_2) = 2\). For the Cartesian product of \(k > 2\) directed cycles, we establish new sufficient conditions for the existence of an antidirected Hamilton cycle.