We present the results of an investigation of the local escape velocity (Vesc) - line strength index relationship for 48 early-type galaxies from the SAURON sample, the first such study based on a large sample of galaxies with both detailed integral field observations and extensive dynamical modelling. Values of Vesc are computed using multi-Gaussian expansion (MGE) photometric fitting and axisymmetric, anisotropic Jeans' dynamical modelling simultaneously on Hubble Space Telescope and ground-based images. We determine line strengths and escape velocities at multiple radii within each galaxy, allowing an investigation of the correlation within individual galaxies as well as amongst galaxies. We find a tight correlation between Vesc and the line-strength indices. For Mgb, we find that this correlation exists not only between different galaxies but also inside individual galaxies - it is both a local and global correlation. The Mgb-Vesc relation has the form: log(Mgb/4Å) = (0.32 +/- 0.03) log(Vesc/500km s-1) - (0.031 +/- 0.007) with an rms scatter σ = 0.033. The relation within individual galaxies has the same slope and offset as the global relation to a good level of agreement, though there is significant intrinsic scatter in the local gradients. We transform our line strength index measurements to the single stellar population (SSP) equivalent ages (t), metallicity ([Z/H]) and enhancement ([α/Fe]) and carry out a principal component analysis of our SSP and Vesc data. We find that in this four-dimensional parameter space the galaxies in our sample are to a good approximation confined to a plane, given by log (V esc/500 kms -1) = 0.85 [Z/H] + 0.43 log (t/Gyr) - 0.29. It is surprising that a combination of age and metallicity is conserved; this may indicate a `conspiracy' between age and metallicity or a weakness in the SSP models. How the connection between stellar populations and the gravitational potential, both locally and globally, is preserved as galaxies assemble hierarchically may provide an important constraint on modelling.