We use deep, spatially resolved spectroscopy from the LEGA-C Survey to study radial variations in the stellar population of 17 spectroscopically-selected post-starburst (PSB) galaxies. We use spectral fitting to measure two Lick indices, $H{delta}_A$ and $Fe4383$, and find that, on average, PSB galaxies have radially decreasing $H{delta}_A$ and increasing $Fe4383$ profiles. In contrast, a control sample of quiescent, non-PSB galaxies in the same mass range shows outwardly increasing $H{delta}_A$ and decreasing $Fe4383$. The observed gradients are weak ($approx-0.2$ r{A}/$R_e$), mainly due to seeing convolution. A two-SSP model suggests intrinsic gradients are as strong as observed in local PSB galaxies ($approx -0.8$ r{A}$/R_e$). We interpret these results in terms of inside-out growth (for the bulk of the quiescent population) vs star formation occurring last in the centre (for PSB galaxies). At $zapprox0.8$, central starbursts are often the result of gas-rich mergers, as evidenced by the high fraction of PSB galaxies with disturbed morphologies and tidal features (40%). Our results provide additional evidence for multiple paths to quiescence: a standard path, associated with inside-out disc formation and with gradually decreasing star-formation activity, without fundamental structural transformation, and a fast path, associated with centrally-concentrated starbursts, leaving an inverse age gradient and smaller half-light radius.
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