  A rearrangement of $n$ independent uniform $[0,1]$ random variables is a
sequence of $n$ random variables $Y_1,...,Y_n$ whose vector of order statistics
has the same distribution as that for the $n$ uniforms. We consider
rearrangements satisfying the strong rank independence condition, that the rank
of $Y_k$ among $Y_1,...,Y_k$ is independent of the values of $Y_1,...,Y_{k-1}$,
for $k=1,...,n$. Nontrivial examples of such rearrangements are the travellers'
processes defined by Gnedin and Krengel. We show that these are the only
examples when $n=2$, and when certain restrictive assumptions hold for $n\geq
3$; we also construct a new class of examples of such rearrangements for which
the restrictive assumptions do not hold.
