  Consider a permutation p to be any finite list of distinct positive integers.
A statistic is a function St whose domain is all permutations. Let S(p,q) be
the set of shuffles of two disjoint permutations p and q. We say that St is
shuffle compatible if the distribution of St over S(p,q) depends only on St(p),
St(q), and the lengths of p and q. This notion is implicit in Stanley's work on
P-partitions and was first explicitly studied by Gessel and Zhuang. 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. The purpose of this paper is to define and study
cyclic shuffle compatibility. In particular, we show how one can lift shuffle
compatibility results for (linear) permutations to cyclic ones. We then apply
this result to cyclic descents and cyclic peaks. We also discuss the problem of
finding a cyclic analogue of the major index.
