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From this, we can then calculate the velocity difference of the particle and B as observed by A. This velocity is not what is observed by B, that would be u'.

Now have u' to the left in B (it is now implicitly "negative").

Once again calculating the velocity difference in A to find the velocity that A observes the particle to have with respect to B, I find that its magnitude is not equal to that of the first case where the particle was moving to the right in B.

I can see why this is the case mathematically, but it doesn't seem intuitive. That is, there is only a change in direction of u' in B, but A observes two different velocities with respect to B.

Can anyone offer me some intuition on this? I have a feeling it has to do with clocks not being synchronized from frame to frame.

Thanks.