There are intervals, and there are intervals.

If we put our yard stick zero point at one end
of a piece of paper and read off the coordinate
at the other end of the paper, we have a good
measure of the paper's length, a Ratio Scale
measure. [Absolute temperature scales are ratio
scale.]

If instead we put the one end of the paper at the
one inch mark (or the zero end of the stick one
inch 'into' the length of the paper) we get measures
that are one inch off the true, ratio scale length.

The two messed up measures are still intervals,
but they are Interval Scale measures. [Household
temperature scales are interval scale, which is
why your physics and chemistry professors won't
let you use them without first converting to the
ratio scale absolute temperatures.)

t'=t/g and x'=x/g represent ratio scale measures,
given that t and x were ratio scalae to start with.

t'=t.z'+t/g and t'=t/g-vx/gcc are both interval 
scale measures, even given a good ratio scale t
and a good ratio scale x.

x'=x.z'+x/g and x'=x/g-vt/g are both interval 
scale measures, even given a good ratio scale x
and a good ratio scale t.

Look for the "(SR) Lorentz t', x' = degraded measures"
document soon at a newsgroup near you.