  A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in
which each alphabet letter appears exactly twice. Such words arise naturally in
the study of topology, graph theory, and combinatorics. Recently, double
occurrence words have been used for studying DNA recombination events. We
develop formulas for counting and enumerating several elementary classes of
double occurrence words such as palindromic, irreducible, and
strongly-irreducible words.
