= Time reversal of the squared diffusion-residual action
For a real scalar height field, <integration by parts> gives
$$
(h_t-\nu\nabla^2h)^2=h_t^2+\nu^2(\nabla^2h)^2+\nu\partial_t|\nabla h|^2-2\nu\nabla\cdot(h_t\nabla h).
$$
The bulk expression is even under <time reversal>. The original density is not pointwise even, but changes by a <total divergence>; its stationary equation is time-reversal invariant. The first-order <diffusion equation> selected by zero residual is not invariant under ordinary time reversal at fixed positive $\nu$. Boundary conditions or the probabilistic interpretation can distinguish the two time directions even when the higher-order stationary equation does not.
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