Symmetry between inertial reference frames So my textbook says the following - roughly translated - in the context of Special Relativity:

"Assume we have two observers, A and B, moving relative to each other. Observer A measures a velocity v for observer B. Because of the symmetry of the situation, observer B measures the same velocity for observer A. If this isn't clear, note that A and B can be replaced by one another. If A and B measured a different velocity, then either of the two would be in a 'special reference frame'. This is in contradiction with the postulates of special relativity."

Alright, so I get this intuitively, of course. However, I can't follow their line of reasoning. How would we know that the laws of physics are different for A and B, if they don't measure the same velocity? Velocity is already relative, so how do we know for sure there would have to be a different set of laws for, say, A to get this result?
I'm familiar with the following argument: We've already deduced that a light clock yields different time intervals between its ticks for different observers. Assume there'd be another 'invariant' clock that would tick the same for any observer (a supposed 'universal time' clock). Then any observer would be able to measure their absolute speed, by comparing the time interval measured by the universal clock and a licht clock (that are assumed to not be movingg relative to one another). This is a contradiction, for velocity can't be absolute by our postulates.
Is it perhaps possible to give the same kind of argument with the symmetry problem? Would it somehow be possible for either of the observers to deduce an 'absolute velocity', if they don't measure the same velocity (which would yield the desired contradiction).
I'm hoping someone could help me out with either of the arguments!
 A: Alright, I got a more intuitive/physical answer (instead of just plugging in values in Galileo/Lorentz transformations!).
We are going to assume two things: object A and object B are moving relative to each other (acceleration=0). Now, we can fix object A and make A face in the direction of B. Assume A tells us that B moves with speed v. Now, we maintain a completely identical situation if we fix B instead of A. In that case we have an object (B, in this case) that is facing another object that moves relative to it [in the same way as above]. This is literally the same situation as above, so if A gives us a value v for B, then B must give us the same value for A.
In short: 
Situation I: An object (A) is moving relative to another (B).
Situation II: An object (B) is moving relative to another (A), in the same way.
Conclusion: Those situations are identical, so any results concocted in situation I must also apply to situation II. 
I know this is an overly cumbersome explanation, but this is the only way I believe I truly get it. :)
