Reconfigurable networks are a novel communication paradigm in which the pattern of connectivity between hosts varies rapidly over time. Prior theoretical work explored the inherent tradeoffs between throughput (or, hop-count) and latency, and showed the existence of infinitely many Pareto-optimal designs as the network size tends to infinity. Existing Pareto-optimal designs use a connection schedule which is fine-tuned to the desired hop-count $h$, permitting lower latency as $h$ increases. However, in reality datacenter workloads contain a mix of low-latency and high-latency requests. Using a connection schedule fine-tuned for one request type leads to inefficiencies when serving other types. A more flexible and efficient alternative is a {\em universal schedule}, a single connection schedule capable of attaining many Pareto-optimal tradeoff points simultaneously, merely by varying the choice of routing paths. In this work we present the first universal schedules for oblivious routing. Our constructions yield universal schedules which are near-optimal for all possible hop-counts $h$. The key technical idea is to specialize to a type of connection schedule based on cyclic permutations and to develop a novel Fourier-analytic method for analyzing randomized routing on these connection schedules. We first show that a uniformly random connection schedule suffices with multiplicative error in throughput, and latency optimal up to a $\log N$ factor. We then show that a more carefully designed random connection schedule suffices with additive error in throughput, but improved latency optimal up to only constant factors. Finally, we show that our first randomized construction can be made deterministic using a derandomized version of the Lovett-Meka discrepancy minimization algorithm to obtain the same result.