(btw, on this forum you have to type [quote] and [/quote] instead of <quote> and </quote> )
Relative simply means that there is a symmetry.
Space (either Galilean or Einsteinian) has translational symmetry, which means that everything behaves the same as seen from two different positions (at the same velocity).
And they have momental symmetry (I forget the correct name
), which means that everything behaves the same as seen from two different velocities.
And rotational symmetry …
Each type of symmetry has a current associated with it (the Noether current, named after female mathematician Emmy Noether), and a conservation law for that current.
A space without symmetries wouldn't be a mathematical space in any sense!
So yes, the "hidden" dimensions do have extra symmetries (either among themselves, or shared with the "public" dimensions), and each produces its own conservation law - and one group-theory problem of string theory is to work out what the symmetries are!