  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.
