  Analogues of 1-shuffle elements for complex reflection groups of type
$G(m,1,n)$ are introduced. A geometric interpretation for $G(m,1,n)$ in terms
of rotational permutations of polygonal cards is given. We compute the
eigenvalues, and their multiplicities, of the 1-shuffle element in the algebra
of the group $G(m,1,n)$. Considering shuffling as a random walk on the group
$G(m,1,n)$, we estimate the rate of convergence to randomness of the
corresponding Markov chain. We report on the spectrum of the 1-shuffle analogue
in the cyclotomic Hecke algebra $H(m,1,n)$ for $m=2$ and small $n$.
