  The card-cyclic-to-random shuffle is the card shuffle where the $n$ cards are
labeled $1,\ldots,n$ according to their starting positions. Then the cards are
mixed by first picking card $1$ from the deck and reinserting it at a uniformly
random position, then repeating for card $2$, then for card $3$ and so on until
all cards have been reinserted in this way. Then the procedure starts over
again, by first picking the card with label $1$ and reinserting, and so on.
Morris, Ning and Peres \cite{MNP} recently showed that the order of the number
of shuffles needed to mix the deck in this way is $n\log n$. In the present
paper, we consider a variant of this shuffle with relabeling, i.e.\ a shuffle
that differs from the above in that after one round, i.e.\ after all cards have
been reinserted once, we relabel the cards according to the positions in the
deck that they now have. The relabeling is then repeated after each round of
shuffling. It is shown that even in this case, the correct order of mixing is
$n\log n$.
