  We establish that, for every hyperbolic orbifold of type (2, q, $\infty$) and
for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent
bundle is left-handed. This implies that the link formed by every collection of
periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii)
is a fibered link. We also prove similar results for the torus with any flat
metric. Besides, we observe that the natural extension of the conjecture to
arbitrary hyperbolic surfaces (with non-trivial homology) is false.
