What's the geometry of the double coset space?

Question

Let $G$ be a Lie group and $K$ a compact subgroup. The double closet space $K\backslash G/K$ is important in harmonic analysis of semisimple Lie groups. How should I think about this space? $G/K$ is a nice manifold, and $G\rightarrow G/K$ is a nice principal $K$ bundle. Is there any kind of similar geometry for $G\rightarrow K\backslash G/K$? I know if $K$ is not compact, the double closet space can be quite bad, so clearly compactness would have to play some role.