  We measure the intrinsic shapes and alignments of the dwarf spheroidal (dSph)
galaxies of the Local Group. We find the dSphs of the Milky Way are
intrinsically flatter (mean intrinsic ellipticity $\mu_E\sim0.6$) than those of
M31 ($\mu_E\sim0.5$) and that the classical Milky Way dSphs
($M_V<-8.5\,\mathrm{mag}$) are rounder ($\mu_E\sim0.5$) than the ultrafaints
($\mu_E\sim0.65$) whilst in M31 the shapes of the classical and ultrafaint
dSphs are very similar. The M31 dSphs are preferentially radially aligned with
a dispersion of $\sim45\deg$. This signal is driven by the ultrafaint
population whilst the classical M31 dSphs are consistent with a random
orientation. We compare our results to the Aquarius mock stellar catalogues of
Lowing et al. and find the subhalo radial alignment distribution matches the
Local Group dSphs results, whilst the Aquarius intrinsic ellipticities are
significantly smaller than the data ($\Delta\langle E\rangle\approx0.4$). We
provide evidence that the major axes of the Milky Way satellites lie within a
preferential plane with normal vector pointing towards $(\ell,b)=(127,5)\deg$.
We associate this preferred direction with the Vast Polar Orbital structure
although their respective great circles are offset by $\sim30\deg$. No signal
in the alignments of the major axes is found in M31, suggesting that the Great
Plane of Satellites is formed from recent accretion or chance alignment.
Finally, we provide predictions for the discrepancy between the velocity
dispersion versus scale radius distributions for the Milky Way and M31
populations and demonstrate that the projection effect from viewing similar
populations from two different locations does not account for the discrepancy
which is probably caused by increased tidal disruption in M31.
