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, HδA and Fe4383, and find that, on average, PSB galaxies have radially decreasing HδA and increasing Fe4383 profiles. In contrast, a control sample of quiescent, non-PSB galaxies in the same mass range shows outwardly increasing HδA and decreasing Fe4383. The observed gradients are weak (≈−0.2Å/Re), mainly due to seeing convolution. A two-SSP model suggests intrinsic gradients are as strong as observed in local PSB galaxies (≈−0.8Å/Re). 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 z ≈ 0.8, 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.