the problem is easily solved by calculating the crossproduct ot tangents of the ingoing and of the outgoing arc; if that vector points away from the sphere's center, then the spherical polygon is oriented counter clockwise (when looking from the outside towards the sphere's center).
Another, even simpler solution of the problem is the following:
under the assumption that the sphere is centered at the origin, all points on the sphere have equal distance from the origin and can also be interpreted as vectors of equal norm.
Lets further assume we are given three linearly independent vectors $$u,v,w\in\mathbb{R}^3\wedge \|u\|_2=\|v\|_2=\|w\|_2$$ of equal length, then the sequence $(u,v,w)$ is left turn on the sphere (and thus indicates counter clockwise traversal) iff $$((v-u)\times(w-v))^Tv\gt0$$