High-magnetic-field radio pulsars are important transition objects for understanding the connection between magnetars and conventional radio pulsars. We present a detailed study of the young radio pulsar J1119-6127, which has a characteristic age of 1900yr and a spin-down-inferred magnetic field of 4.1e13G, and its associated supernova remnant G292.2-0.5, using deep XMM-Newton and Chandra X-ray Observatory exposures of over 120ks from each telescope. The pulsar emission shows strong modulation

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... elow 2.5keV, with a single-peaked profile and a large pulsed fraction of 0.48+/-0.12. Employing a magnetic, partially ionized hydrogen atmosphere model, we find that the observed pulse profile can be produced by a single hot spot of temperature 0.13keV covering about one third of the stellar surface, and we place an upper limit of 0.08keV for an antipodal hot spot with the same area. The nonuniform surface temperature distribution could be the result of anisotropic heat conduction under a strong magnetic field, and a single-peaked profile seems common among high-B radio pulsars. For the associated remnant G292.2-0.5, its large diameter could be attributed to fast expansion in a low-density wind cavity, likely formed by a Wolf-Rayet progenitor, similar to two other high-B radio pulsars.