In this paper we introduce and study the \textsc{StreamingCycles} problem, a random order streaming version of the Boolean Hidden Hypermatching problem that has been instrumental in streaming lower bounds over the past decade. In this problem the edges of a graph $G$, comprising $n/\ell$ disjoint length-$\ell$ cycles on $n$ vertices, are partitioned randomly among $n$ players. Every edge is annotated with an independent uniformly random bit, and the players' task is to output the parity of some cycle in $G$ after one round of sequential communication. Our main result is an $\ell^{Ω(\ell)}$ lower bound on the communication complexity of \textsc{StreamingCycles}, which is tight up to constant factors in $\ell$. Applications of our lower bound for \textsc{StreamingCycles} include an essentially tight lower bound for component collection in (almost) random order graph streams, making progress towards a conjecture of Peng and Sohler [SODA'18] and the first exponential space lower bounds for random walk generation.