Topological insulators - Surface states have a phase?

When I look at the circle of the Dirac cone around the Dirac point of, let's say, $Bi_2Se_3$, then the electron winds around and it is true that it goes from momentum $-k$ and spin-up to $+k$ and spin-down. Now how can I use this fact to show that the Berry phase of $\pi$ arises?

When the electron completes one circle it did go from $k$ to $k$ and from spin-up to spin-up. So basically nothing changed...?

Picture that shows the Dirac cone in the Brillouin zone and the spin on the left hand side: