  We investigate the connection between the morphology and internal kinematics
of the stellar component of central galaxies with mass $M_\star > {10}^{9.5}
{\rm M}_\odot$ in the EAGLE simulations. We compare several kinematic
diagnostics commonly used to describe simulated galaxies, and find good
consistency between them. We model the structure of galaxies as ellipsoids and
quantify their morphology via the ratios of their principal axes. We show that
the differentiation of blue star-forming and red quiescent galaxies using
morphological diagnostics can be achieved with similar efficacy to the use of
kinematical diagnostics, but only if one is able to measure both the flattening
and the triaxiality of the galaxy. Flattened oblate galaxies exhibit greater
rotational support than their spheroidal counterparts, but there is significant
scatter in the relationship between morphological and kinematical diagnostics,
such that kinematically-similar galaxies can exhibit a broad range of
morphologies. The scatter in the relationship between the flattening and the
ratio of the rotation and dispersion velocities ($v/\sigma$) correlates
strongly with the anisotropy of the stellar velocity dispersion: at fixed
$v/\sigma$, flatter galaxies exhibit greater dispersion in the plane defined by
the intermediate and major axes than along the minor axis, indicating that the
morphology of simulated galaxies is influenced significantly by the structure
of their velocity dispersion. The simulations reveal that this anisotropy
correlates with the intrinsic morphology of the galaxy's inner dark matter
halo, i.e. the halo's morphology that emerges in the absence of dissipative
baryonic physics. This implies the existence of a causal relationship between
the morphologies of galaxies and that of their host dark matter haloes.
