Relative speed conservation in Elastic Collisions (One dimensional)

Why is relative speed conserved in elastic collisions?

The algebraic derivation of the equation: $u_1 - u_2 = v_2 - v_1$ or $|u_1 - u_2| = |v_1 - v_2|$ is relatively (no pun intended) straightforward. However I'm asking if anyone can think of a good intutive explanation for why the relative speeds of these particles are conserved. It seems bizarre to me that this would fall out of it being simply an elastic collision. Thanks.