  Chung and Graham define quasirandom subsets of $\mathbb{Z}_n$ to be those
with any one of a large collection of equivalent random-like properties. We
weaken their definition and call a subset of $\mathbb{Z}_n$ $\epsilon$-balanced
if its discrepancy on each interval is bounded by $\epsilon n$. A quasirandom
permutation, then, is one which maps each interval to a highly balanced set. In
the spirit of previous studies of quasirandomness, we exhibit several
random-like properties which are equivalent to this one, including the property
of containing (approximately) the expected number of subsequences of each
order-type. We provide a few applications of these results, present a
construction for a family of strongly quasirandom permutations, and prove that
this construction is essentially optimal, using a result of W. Schmidt on the
discrepancy of sequences of real numbers.
