For even $n>0$, let $K_n$ be the complete graph with vertices $v_0, v_1, \cdots, v_{n-1}$. An edge $v_iv_j$ is called odd or even accordingly as $|i-j|$ is odd or even. An odd(even) 1-factor of $K_n$ is a 1-factor of $K_n$ whose edges are all odd(even). The Dihedral group $D_n$ acts on $K_n$ naturally, and this action induces an action of $D_n$ on the family of all 1-factors of $K_n$. In this paper, by applying Burnside's lemma, we calculate the number of the equivalence classes of odd(even) 1-factors under the action of $D_n$.