What do you mean with usually? Spacetime (i.e. the object of discussion in relativity) is by definition not positive definite. You said it yourself that space and time require different signs. But they are still interchangeable - without changing physics. On top of that, all that a Lorentz transformation (at least for boosts) does is rotate (hyperbolically) space and time into each other. Seeing space and time as two different things rather than interchangeable components of a more fundamental object is not in line with relativity.
