Take an equator on the two sphere $S^2$ and parametrize it by arc-length obtaining a closed loop $\alpha: S^1 \to S^2$. The curve $(\alpha,\alpha'):S^1 \to T^1S^2$ in the unit tangent bundle of $S^2$ is homotopically non-trivial.

However if you consider the concatenation $\beta$ of two copies of $\alpha$, you can take one copy and turn it about a diameter passing through two points of $\alpha$ so that it is now a copy of $\alpha$ but traversed in the opposite sense. In other words, the curve $(\beta,\beta') \in T^1S^2$ is homotopically trivial.

Does this occur on other compact orientable surfaces?

On the torus the answer is no.

What about on the double-torus (which is a hyperbolic surface)?

Also, does anybody have references for the above statements about the sphere and the torus? In particular, are they correct?

I've come across these problems while trying to picture the lift of a general geodesic flow to the universal covering space of $T^1M$ where $M$ is a compact orientable surface (in particular, what do homotopically trivial closed geodesics look like?).