  Pulsar timing arrays seek and study gravitational waves (GWs) through the
angular two-point correlation function of timing residuals they induce in
pulsars. The two-point correlation function induced by the standard
transverse-traceless GWs is the famous Hellings-Downs curve, a function only of
the angle between the two pulsars. Additional polarization modes
(vector/scalar) that may arise in alternative-gravity theories have different
angular correlation functions. Furthermore, anisotropy, linear, or circular
polarization in the stochastic GW background gives rise to additional structure
in the two-point correlation function that cannot be written simply in terms of
the angular separation of the two pulsars. In this paper, we provide a simple
formula for the most general two-point correlation function -- or overlap
reduction function (ORF) -- for a gravitational-wave background with an
arbitrary polarization state, possibly containing anisotropies in its intensity
and polarization (linear/circular). We provide specific expressions for the
ORFs sourced by the general-relativistic transverse-traceless GW modes as well
as vector (or spin-1) modes that may arise in alternative-gravity theories.
