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I show that it is possible to formulate the Relativity postulates in a waythat does not lead to inconsistencies in the case of space-times whoseshort-distance structure is governed by an observer-independent length scale.The consistency of these postulates proves incorrect the expectation thatmodifications of the rules of kinematics involving the Planck length wouldnecessarily require the introduction of a preferred class of inertialobservers. In particular, it is possible for every inertial observer to agreeon physical laws supporting deformed dispersion relations of the type $E^2- c^2p^2- c^4 m^2 + f(E,p,m;L_p)=0$, at least for certain types of $f$.Comment: Same formulas and results as in 1st version, but a change of notation  is introduced in order to clarify that the studied illustrative example is  consistent with the R.P. for both choices of the overall sign. 1 ref added  and 2 refs upgraded. Some rewording of the text in Sec5, and addition of an  analogy with background fields in ordinary electromagnetism which I use to  illustrate difference between space-times with an observer-independent Lp,  and space-times in which Lp is introduced without modifications of Special  Relativit

RELATIVITY IN SPACETIMES WITH SHORT-DISTANCE STRUCTURE GOVERNED BY AN OBSERVER-INDEPENDENT (PLANCKIAN) LENGTH SCALE