Although the importance of clock synchronization for relativity is discussed from time to time in the educational literature, the fact that different synchronization conventions imply different caordinatizations of spacetime with ensuing changes of the form of possibly all coordinate-dependent quantities, has neither entered textbooks nor undergraduate physics education. As a consequence, lhere is a widespread belief among students that the familiar form ofcoordinatedependent quantities like the measured velocity of light, the Lorentz transformation between two observers, 'addition of velocities', 'time dilation', 'length contraction', 'E = d y ' , which they assume under the standard clock synchronization, is relativity proper. In order to demonstrate that this is by no means so, this paper studies the consequences of a non-standard synchronization, and it is shown that drastic changes in the appearance of all these quantities are thus induced. For example, the phrases 'moving clocks go slow', and 'simultaneity is relative', which are usually considered as intrinsic features of relativity, turn out to be no longer true, whereas all coordinate-independent quantities remain of course indifferent to such a change in coordinatization. This demonstration is all the more thought-provoking as the synchronization adapted is by no means far-fetched. but is the procedure used in everyday life (' ...at the sound of the last tone, it will be 12 o'clock'). And although Einstein's standard convention of clock synchronization enjoys distinct advantages over the 'everyday' method, the message clearly conveyed is that in the teaching of elementary relativity much more stress should be laid on the intrinsic (coordinate-independent) features of spacetime.