The twist two contribution in the operator product expansion of $\phi_1(x_1) \phi_2(x_2)$ for a pair of globally conformal invariant, scalar fields of equal scaling dimension d in four space-time dimensions is a field $V_1(x_1,x_2)$ which is harmonic in both variables. It is demonstrated that the Huygens bilocality of $V_1$ can be equivalently characterized by a ``single-pole property'' concerning the pole structure of the (rational) correlation functions involving the product $\phi_1(x_1) \phi_2(x_2)$. This property is established for the dimension d=2 of $\phi_1$, $\phi_2$. As an application we prove that any GCI scalar field of conformal dimension 2 (in four space-time dimensions) can be written as a (possibly infinite) superposition of products of free massless fields.