The traditional presentation of special relativity is made from a rupture with previous ideas, such as the notion of absolute motion, emphasizing the antagonism of the Lorentz-Poincaré's views and Einstein's ideas. However, a weaker formulation of the postulates allows to recover all the mathematical results from Einstein's special relativity and reveals that both viewpoints are merely different perspectives of one and the same theory. The apparent contradiction simply stems from different procedures for clock "synchronization," associated with different choices of the coordinates used to describe the physical world. Even very fundamental claims, such as the constancy of the speed of light, relativity of simultaneity and relativity of time dilation, are seen to be no more than a consequence of a misleading language adopted in the description of the physical reality, which confuses clock rhythms with clock time readings. Indeed, the latter depend on the "synchronization" adopted, whereas the former do not. As such, these supposedly fundamental claims are not essential aspects of the theory, as reality is not altered by a mere change of coordinates. The relation between the rhythms of clocks in relative motion is derived with generality. This relation, which is not the standard textbook expression, markedly exposes the indeterminacy of special relativity, connected with the lack of knowledge of the value of the one-way speed of light. Moreover, the theory does not collapse and remains valid if some day the one-way speed of light is truly measured and the indeterminacy is removed. It is further shown that the slow transport method of "synchronization" cannot be seen as distinct from Einstein's procedure.