We use the Galaxies–Intergalactic Medium Interaction Calculation (GIMIC) suite of cosmological hydrodynamical simulations to study the global structure and kinematics of stellar spheroids of Milky Way mass disc galaxies. Font et al. have recently demonstrated that these simulations are able to successfully reproduce the satellite luminosity functions and the metallicity and surface brightness profiles of the spheroids of the Milky Way and M31. A key to the success of the simulations is a significant contribution to the spheroid from stars that formed in situ. While the outer halo is dominated by accreted stars, stars formed in the main progenitor of the galaxy dominate at r≲ 30 kpc. In the present study, we show that this component was primarily formed in a protodisc at high redshift and was subsequently liberated from the disc by dynamical heating associated with mass accretion. As a consequence of its origin, the in situ component of the spheroid has different kinematics (namely net prograde rotation with respect to the disc) than that of the spheroid component built from the disruption of satellites. In addition, the in situ component has a flattened distribution, which is due in part to its rotation. We make comparisons with measurements of the shape and kinematics of local galaxies, including the Milky Way and M31, and stacked observations of more distant galaxies. We find that the simulated disc galaxies have spheroids of the correct shape (oblate with a median axial ratio of ∼0.6 at radii of ≲30 kpc, but note there is significant system-to-system scatter in this quantity) and that the kinematics show evidence for two components (due to in situ versus accreted), as observed. Our findings therefore add considerable weight to the importance of dissipative processes in the formation of stellar haloes and to the notion of a ‘dual stellar halo’.