In a purely cold dark matter universe, the initial matter power spectrum and its subsequent gravitational growth contain no special mass- or time-scales, and so neither do the emergent population statistics of internal dark matter (DM) halo properties. Using 1.5 million halos from three IllustrisTNG realizations of a LambdaCDM universe, we show that galaxy formation physics drives non-monotonic features (wiggles) into DM property statistics across six decades in halo mass, from dwarf galaxies to galaxy clusters. We characterize these features by extracting the halo mass-dependent statistics of five DM halo properties -- velocity dispersion, NFW concentration, density- and velocity-space shapes, and formation time -- using kernel-localized linear regression (KLLR). Comparing precise estimates of normalizations, slopes, and covariances between realizations with and without galaxy formation, we find systematic deviations across all mass-scales, with maximum deviations of 25% at the Milky-Way mass of 1e12 Msun. The mass-dependence of the wiggles is set by the interplay between different cooling and feedback mechanisms, and we discuss its observational implications. The property covariances depend strongly on halo mass and physics treatment, but the correlations are mostly robust. Using multivariate KLLR and interpretable machine learning, we show the halo concentration and velocity-space shape are principal contributors, at different mass, to the velocity dispersion variance. Statistics of mass accretion rate and DM surface pressure energy are provided in an appendix. We publicly release halo property catalogs and KLLR parameters for the TNG runs at twenty epochs up to z = 12.
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