We present a comprehensive statistical analysis of Swift X-ray light curves of gamma-ray bursts (GRBs) collecting data from more than 650 GRBs discovered by Swift and other facilities. The unprecedented sample size allows us to constrain the rest-frame X-ray properties of GRBs from a statistical perspective, with particular reference to intrinsic time-scales and the energetics of the different light-curve phases in a common rest-frame 0.3-30 keV energy band. Temporal variability episodes are also studied and their properties constrained. Two fundamental questions drive this effort: (i) Does the X-ray emission retain any kind of 'memory' of the prompt γ-ray phase? (ii) Where is the dividing line between long and short GRB X-ray properties? We show that short GRBs decay faster, are less luminous and less energetic than long GRBs in the X-rays, but are interestingly characterized by similar intrinsic absorption. We furthermore reveal the existence of a number of statistically significant relations that link the X-ray to prompt γ-ray parameters in long GRBs; short GRBs are outliers of the majority of these two-parameter relations. However and more importantly, we report on the existence of a universal three-parameter scaling that links the X-ray and the γ-ray energy to the prompt spectral peak energy of both long and short GRBs: E X, iso∝E 1.00 ± 0.06 γ, iso/E 0.60 ± 0.10 pk.