  In this paper we study the twistor space $Z$ of an oriented Riemannian
four-manifold $M$ using the moving frame approach, focusing, in particular, on
the Einstein, non-self-dual setting. We prove that any general first-order
linear condition on the almost complex structures of $Z$ forces the underlying
manifold $M$ to be self-dual, also recovering most of the known related
rigidity results. Thus, we are naturally lead to consider first-order quadratic
conditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor
space of an Einstein four-manifold bears a resemblance, in a suitable sense, to
a nearly K\"ahler manifold.
