So, this is how it is typically done. We have a reference frame $S$ and the coordinates $(x,t)$ of an event, $E$ (or collection of events which we call world-lines) are mapped onto a 2-dimensional physical paper. (Typically $x$ is horizontal axis and $t$ is vertical axis and the two axis are orthogonal)
Now we analyse the same "event-space" (i.e. event $E$ and the world-lines etc) with respect to another reference frame $S'$ which is moving relative to $S$. We need new set of co-ordinates for $S'$ to "measure" the event-space (i.e to mark an event on the piece of paper with respect to $S'$). We get the new co-ordinates for $x'$ and $t'$ which are just "rotation" (in opp directions) of original axis (rotation depends on relative speed between $S$ and $S'$) and $x'$ and $t'$ are not orthogonal.
My question: We in the above construction implicitly assume that we do not need to "re-draw" the events and world-lines on the paper. The events and world-lines we marked in the $S$ co-ordinate system will continue to be on the same "physical" point (or line if world-line) on the paper in the new co-ordinate system of $S'$.
Why is above assumption acceptable? Why are we guaranteed that by mere rotation of the co-ordinate axis we will get the co-ordinate system for $S'$ in which the physical layout on the sheet of paper of the event-space remains same as it was in $S$ co-ordinate system? Is this unique to special-relativity?