Classifying Rotationally-Closed Languages Having Greedy Universal Cycles

Let $\mathbf{\text{T}}(n,k)$ be the set of strings of length $n$ over the alphabet $\mathrm{\Sigma }=\{1,2,\dots ,k\}$. A universal cycle for $\mathbf{\text{T}}(n,k)$ can be constructed using a greedy algorithm: start with the string $k^n$, and continually append the least symbol possible without repeating a substring of length $n$. This construction also creates universal cycles for some subsets $\mathbf{\text{S}}\subseteq \mathbf{\text{T}}(n,k)$; we will classify all such subsets that are closed under rotations.