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 galaxys inner dark matter halo, i.e. the halos 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.
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