  A centrosymmetric permutation is one which is invariant under the
reverse-complement operation, or equivalently one whose associated standard
Young tableaux under the Robinson-Schensted algorithm are both invariant under
the Schutzenberger involution. In this paper, we characterize the set of
permutations avoiding 1243 and 2143 whose images under the reverse-complement
mapping also avoid these patterns. We also characterize in a simple manner the
corresponding Schroder paths under a bijection of Egge and Mansour. We then use
these results to enumerate centrosymmetric permutations avoiding the patterns
1243 and 2143. In a similar manner, centrosymmetric involutions avoiding these
same patterns are shown to be enumerated by the Pell numbers.
