  The large separation in the low radial order regime is considered as a highly
valuable observable to derive mean densities of $\delta$ Scuti stars, due to
its independence with rotation. Up to now, theoretical studies of this
$\Delta\nu$-${\bar \rho}$ relation have been limited to 1D non-rotating models,
and 2D pseudo-evolutionary models. The present work aims at completing this
scenario by investigating quantitatively the impact of rotation in this
relation on a large grid of 1D asteroseismic models representative of $\delta$
Scuti stars. These include rotation effects on both the stellar evolution and
the interaction with pulsation. This allowed us to compute the stellar
deformation, get the polar and equatorial radii, and correct the stellar mean
densities. We found that the new $\Delta\nu$-${\bar \rho}$ relation for
rotating models is compatible with previous works. We explained the dispersion
of the points around the linear fits as caused mainly by the distribution of
the stellar mass, and partially by the evolutionary stage. The new fit is found
to be close to the previous theoretical studies for lower masses
($1.3-1.81\,\mathrm{M}_{\odot}$). However, the opposite holds for the
observations: for the higher masses ($1.81-3\,\mathrm{M}_{\odot}$) the fit is
more compatible with the empirical relation. We applied these results to
characterise the two well-known $\delta$ Scuti stars observed by CoRoT,
HD174936 and HD174966, and compared the physical parameters with those of
previous works. Inclusion of rotation in the modelling causes a tendency
towards greater masses, radii, luminosities and lower density values.
Comparison between $\Delta\nu$ and Gaia's luminosities also allowed us to
constraint the inclination angles and rotational velocities of both stars. The
present results pave the way to systematically constrain the angle of
inclination of $\delta$ Scuti stars
