We report a Dalitz-plot analysis of the charmless hadronic decays of neutral B mesons to K ± π ∓ π 0 . With a sample of ( 231.8 ± 2.6 ) × 10 6 Υ ( 4 S ) → B ¯¯¯ B decays collected by the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC, we measure the magnitudes and phases of the intermediate resonant and nonresonant amplitudes for B 0 and ¯¯¯ B 0 decays and determine the corresponding C P -averaged branching fractions and charge asymmetries. The inclusive branching fraction and C P -violating charge asymmetry are measured to be B ( B 0 → K + π − π 0 ) = ( 35.7 + 2.6 − 1.5 ± 2.2 ) × 10 − 6 and A C P = − 0.030 + 0.045 − 0.051 ± 0.055 where the first errors are statistical and the second systematic. We observe the decay B 0 → K * 0 ( 892 ) π 0 with the branching fraction B ( B 0 → K * 0 ( 892 ) π 0 ) = ( 3.6 + 0.7 − 0.8 ± 0.4 ) × 10 − 6 . This measurement differs from zero by 5.6 standard deviations (including the systematic uncertainties). The selected sample also contains B 0 → ¯¯¯ D 0 π 0 decays where ¯¯¯ D 0 → K + π − , and we measure B ( B 0 → ¯¯¯ D 0 π 0 ) = ( 2.93 ± 0.17 ± 0.18 ) × 10 − 4 .