Consider a metric space $(\mathcal{X}, d)$ and four points $x, y, z, u \in \mathcal{X}$.
When $\mathcal{X} = \mathbb{R}$ and $d(x,x') = |x - x'|$, then the metric has the following interesting property. There is a perfect matching between the terms $$d(x,y) + d(y,z), \qquad d(x,u) + d(u,z), \qquad d(x,z) + 2 d(y,u)$$ and the terms $$d(x,y) + d(y,u) + d(u,z), \qquad d(x,u) + d(u,y) + d(y,z), \qquad d(x,z)$$ such that the matched terms are equal. For example, when $x \leq y \leq u \leq z$ then $$d(x,y)+d(y,z) = d(x,z), \quad d(x,u)+d(u,z) = d(x,z), \quad d(x,y) + d(y,u) + d(u,z) = d(x,z), \quad d(x,u) + d(u,y) + d(y,z) = d(x,z) + 2 d(y,u).$$
Intuitively, there is a certain symmetry between $x,y,z,u$ such that the triangle inequality turns into an equality in certain cases.
I am wondering: Is this a peculiar case of $\mathcal{X} = \mathbb{R}$ and $d(x,x') = |x - x'|$ or is there a more general phenomenon that explains this intriguing behavior?