Homogeneous spaces with a cut point

Question

If a connected homogeneous space has a cut point, then every point is a cut point by homogeneity. But just how strong is this condition? Intuitively and informally, it seems to provide a notion that the space is a continuous line, since every point "looks the same as every other point" and also "bridges different components of the space". The only examples I can think of with this property are $\mathbb{R}$ and two long line constructions, where you either make a homogeneous space which is "long" in only one direction by deleting the endpoint of $\omega_1 \times [0,1)$ or make one which is "long" in both directions by gluing together two copies of $\omega_1 \times [0,1)$ at their endpoints. I guess you could also count the singleton space as a degenerate example, if you consider the empty space to not be connected. Do all such spaces resemble $\mathbb{R}$ in some way? Is it just a coincidence that all of these examples are connected manifolds of dimension 1 (or 0 in the trivial case)? Are there even any other examples beyond these three (or four)?