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The inversion of multitemporal DInSAR and GPS measurements unravels the coseismic and postseismic (afterslip) slip distributions associated with the 2009 M-W 6.3 L'Aquila earthquake and provides insights into the rheological properties and long-term behavior of the responsible structure, the Paganica fault. Well-resolved patches of high postseismic slip (10-20 cm) appear to surround the main coseismic patch (maximum slip approximate to 1 m) through the entire seismogenic layer above the hypocenter without any obvious depth-dependent control. Time series of postseismic displacement are well reproduced by an exponential function with best-fit decay constants in the range of 20-40 days. A sudden discontinuity in the evolution of released postseismic moment at approximate to 130 days after the main shock does not correlate with independent seismological and geodetic data and is attributed to residual noise in the InSAR time series. The data are unable to resolve migration of afterslip along the fault probably because of the time interval (six days) between the main shock and the first radar acquisition. Surface fractures observed along the Paganica fault follow the steepest gradients of postseismic line-of-sight satellite displacements and are consistent with a sudden and delayed failure of the shallow layer in response to upward tapering of slip. The occurrence of afterslip at various levels through the entire seismogenic layer argues against exclusive depth-dependent variations of frictional properties on the fault, supporting the hypothesis of significant horizontal frictional heterogeneities and/or geometrical complexities. We support the hypothesis that such heterogeneities and complexities may be at the origin of the long-term variable behavior suggested by the paleoseismological studies. Rupture of fault patches with dimensions similar to that activated in 2009 appears to have a approximate to 500 year recurrence time interval documented by paleoseismic and historical studies. In addition to that, paleoseismological evidence of large (>0.5 m) coseismic offsets seems to require seismic events, recurring every 1000-2000 years, characterized by (1) multisegment linkage, (2) surface ruptures larger than in 2009, and (3) complete failure of the 2009 coseismic and postseismic patches.