The internal properties of dark matter haloes correlate with the large-scale halo clustering strength at fixed halo mass $-$ an effect known as assembly bias $-$ 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-scale correlations represent two distinct statistical links: one between the internal property and $\alpha$, and the other between $\alpha$ and large-scale ( $>30h^{-1}$Mpc) halo bias $b_1$... We focus on scalar internal properties of haloes related to formation time (concentration $c_{\rm vir}$), shape (mass ellipsoid asphericity $c/a$), velocity dispersion structure (velocity ellipsoid asphericity $c_v/a_v$ and velocity anisotropy $\beta$) and angular momentum (dimensionless spin $\lambda$) in the mass range $8\times10^{11}< M_{\rm vir}/(h^{-1}M_\odot)<5\times10^{14}$. Using conditional correlation coefficients 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 the largest masses. $\textit{Thus, the assembly bias trends $c-b_1$ reflect the two fundamental correlations $c-\alpha$ and $b_1-\alpha$. }$ Our results are unaffected by the exclusion of haloes with recent major merger events or splashback objects, although the latter are distinguished by the fact that $\alpha$ does not explain their assembly bias trends. The overarching importance of $\alpha$ provides a new perspective on the nature of assembly bias of distinct haloes, with potential ramifications for incorporating realistic assembly bias effects into mock catalogs of future large-scale structure surveys and for detecting galaxy assembly bias. (read more)