  Given a set $V$, a subset $S$, and a permutation $\pi$ of $V$, we say that
$\pi$ permutes $S$ if $\pi (S) \cap S = \emptyset$. Given a collection $\cS =
\{V; S_1,\ldots , S_m\}$, where $S_i \subseteq V ~~(i=1,\ldots ,m)$, we say
that $\cS$ is invertible if there is a permutation $\pi$ of $V$ such that $\pi
(S_i) \subseteq V-S_i$. In this paper, we present necessary and sufficient
conditions for the invertibility of a collection and construct a polynomial
algorithm which determines whether a given collection is invertible. For an
arbitrary collection, we give a lower bound for the maximum number of sets that
can be inverted. Finally, we consider the problem of constructing a collection
of sets such that no sub-collection of size three is invertible. Our
constructions of such collections come from solutions to the packing problem
with unbounded block sizes. We prove several new lower and upper bounds for the
packing problem and present a new explicit construction of packing.
