There's a nice answer to this question: Why is the scalar product of two four-vectors Lorentz-invariant? - that explains that a Lorentz transformation is one under which the inner product of two 4-vectors is invariant.
I know that the norm of the difference between two 4-vectors can be interpreted as the spacetime separation between corresponding events, and I understand the reason for invariance of the spacetime interval. But the inner product is more general than norm of difference, so what's the physical reason for saying that the LT should preserve the inner product of 4-vectors?
Alternatively I could phrase my question as follows: is there an explanation why a Lorentz transformation is equivalent to a change of basis? (because the inner product is invariant under a change of basis - the vectors essentially stay the same - but isn't generally invariant under a linear transformation)