  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 $z\approx0.8$, 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.
