The distance distribution of a code is the vector whose $i^\text{th}$ entry is the number of pairs of codewords with distance $i$. We investigate the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of $\mathbb{F}_{q^n}^*$ on an $\mathbb{F}_q$-subspace $U$ of $\mathbb{F}_{q^n}$. We show that for optimal full-length orbit codes the distance distribution depends only on $q,\,n$, and the dimension of $U$. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the distance distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. Finally, we briefly address the distance distribution of a union of optimal full-length orbit codes.