A permutation statistic $\st$ is said to be shuffle-compatible if the distribution of $\st$ over the set of shuffles of two disjoint permutations $\pi$ and $\sigma$ depends only on $\st\pi$, $\st\sigma$, and the lengths of $\pi$ and $\sigma$. Shuffle-compatibility is implicit in Stanley's early work on $P$-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. One of the places where shuffles are useful is in describing the product in the algebra of quasisymmetric functions. Recently Adin, Gessel, Reiner, and Roichman defined an algebra of cyclic quasisymmetric functions where a cyclic version of shuffling comes into play. This dissertation focuses on the study of cyclic shuffle-compatibility. We began by showing a result called the ``lifting lemma,'' which allows one (under certain nice conditions) to prove that a cyclic statistic is cyclic shuffle-compatible from the shuffle-compatibility of a related linear statistic. This lifting lemma can be used to prove the cyclic shuffle-compatibility of all four statistics $\cDes$, $\cdes$, $\cPk$, and $\cpk$. We then developed an algebraic framework for cyclic shuffle-compatibility centered around the notion of cyclic shuffle algebra of a cyclic shuffle-compatible statistic. Using this theory, we provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility. In particular, we developed the theory of enriched toric $[\vec{D}]$-partitions, which provides a characterization of the cyclic shuffle algebra of $\cPk$.
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