  A bounded linear operator $A$ on a Hilbert space is posinormal if there
exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. Posinormality of
$A$ is equivalent to the inclusion of the range of $A$ in the range of its
adjoint $A^*$. Every hyponormal operator is posinormal, as is every invertible
operator. We characterize both the posinormal and coposinormal composition
operators $C_\varphi$ on the Hardy space $H^2$ of the open unit disk
$\mathbb{D}$ when $\varphi$ is a linear-fractional selfmap of $\mathbb{D}$. Our
work reveals that there are composition operators that are both posinormal and
coposinormal yet have powers that fail to be posinormal.
