  An angular analysis of the $B^0 \to K^{*0} e^+ e^-$ decay is performed using
a data sample corresponding to an integrated luminosity of $9~{\rm fb}^{-1}$ of
$pp$ collisions collected with the LHCb experiment. The analysis is conducted
in the very low dielectron mass squared ($q^2$) interval between $0.0008$ and
$0.257~{\rm GeV}^2$, where the rate is dominated by the $B^0\to K^{\ast
0}\gamma$ transition with a virtual photon. The fraction of longitudinal
polarisation of the $K^{\ast 0}$ meson, $F_{\rm L}$, is measured to be $F_{\rm
L} = (4.4 \pm 2.6 \pm 1.4)\%$, where the first uncertainty is statistical and
the second systematic. The $A_{\rm T}^{\rm Re}$ observable, which is related to
the lepton forward-backward asymmetry, is measured to be $A_{\rm T}^{\rm
Re}=-0.06 \pm 0.08 \pm 0.02$. The $A_{\rm T}^{(2)}$ and $A_{\rm T}^{\rm Im}$
transverse asymmetries, which are sensitive to the virtual photon polarisation,
are found to be $A_{\rm T}^{(2)} = 0.11 \pm 0.10 \ \pm 0.02$ and $A_{\rm
T}^{\rm Im} = 0.02 \pm 0.10 \pm 0.01$. The results are consistent with Standard
Model predictions and provide the world's best constraint on the $b\to s\gamma$
photon polarisation.
