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According to wikipedia, we get

Lorentz boost (x direction)

and slightly different formulas for the inverse Lorentz boost

"This "trick" of simply reversing the direction of relative velocity while preserving its magnitude..." - Wikipedia *snap*

You cannot reverse the direction of relative velocity. The whole point is that it is *relative*

Taking two observers A and A' which move at v relative to each other, with A measuring an even e1(x1,t1) and A' measuring e1'(x2,t2) then it SHOULD follow by the equivalence principle,

that if A' was to measure an event e2 with the same values of e1, hence e2'(x1,t1) then A has to necessarily measure this even happening with the SAME values as e1', hence e2(x2,t2).

Only then can we talk about this being equivalent.

But this is NOT the case, as according to wikipedia we have to use different formulas when transforming events from A to A' compared to transforming events from A' to A

x=γ(x'+vt') should be x=γ(x'-vt') instead OR the former for both cases.

What changes is not the velocity sign, since the velocity is relative.

What changes are the signs of the events' coordinates depending on how the observers are facing in relation to each other.

Of course, i might be wrong, but my brain is starting to wobble, so i will have to stop here before serious damage. I might pick up on it later.