So I've started again and when I just use the lighting skybox with the cubemap then the vertical movement is how it's supposed to be. As soon as I put the camera on the CenterAnchor and assign a skybox with the same cubemap then the curved vertical movement happens again.
How did you set up your skybox material? See http://docs.unity3d.com/Manual/HOWTO-UseSkybox.html. Does the built-in default skybox have the same issue? I've tried to reproduce this with a variety of setups, but they all appear to work correctly. It would help if you could share your project or at least the skybox material (and textures) you're using.