  In this paper we consider orbit pairs contributing in the third order of the
spectral form factor in the Hadamard-Gutzwiller model. We prove that periodic
orbits including two 2-encounters in certain structures have partner orbits.
The action differences are estimated at $\ln(1+u_1s_1)(1+u_2s_2)$ with explicit
error bounds, where $(u_1,s_1)$ and $(u_2,s_2)$ are the coordinates of the
piercing points. A new symbolic dynamics for orbit pairs via conjugacy classes
is also provided.
