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The internal properties of dark matter haloes correlate with the large-scalehalo clustering strength at fixed halo mass $-$ an effect known as assemblybias $-$ and are also strongly affected by the local, non-linear cosmic web.Characterising a halo's local web environment by its tidal anisotropy $\alpha$at scales $\sim4$ x the halo radius, we demonstrate that these multi-scalecorrelations represent two distinct statistical links: one between the internalproperty and $\alpha$, and the other between $\alpha$ and large-scale ($>30h^{-1}$Mpc) halo bias $b_1$. We focus on scalar internal properties ofhaloes related to formation time (concentration $c_{\rm vir}$), shape (massellipsoid asphericity $c/a$), velocity dispersion structure (velocity ellipsoidasphericity $c_v/a_v$ and velocity anisotropy $\beta$) and angular momentum(dimensionless spin $\lambda$) in the mass range $8\times10^{11}< M_{\rmvir}/(h^{-1}M_\odot)<5\times10^{14}$. Using conditional correlationcoefficients and other detailed tests, we show that the joint distribution of$\alpha$, $b_1$ and any of the internal properties$c\in\{\beta,c_v/a_v,c/a,c_{\rm vir},\lambda\}$ is consistent with$p(\alpha,b_1,c)\simeq p(\alpha)p(b_1|\alpha)p(c|\alpha)$, at all but thelargest masses. $\textit{Thus, the assembly bias trends $c-b_1$ reflect the twofundamental correlations $c-\alpha$ and $b_1-\alpha$.}$ Our results areunaffected by the exclusion of haloes with recent major merger events orsplashback objects, although the latter are distinguished by the fact that$\alpha$ does not explain their assembly bias trends. The overarchingimportance of $\alpha$ provides a new perspective on the nature of assemblybias of distinct haloes, with potential ramifications for incorporatingrealistic assembly bias effects into mock catalogs of future large-scalestructure surveys and for detecting galaxy assembly bias.

Cosmic web anisotropy is the primary indicator of halo assembly bias