  Recently, values for the Kumar quadrupole deformation parameters of the
nucleus $^{130}$Xe have been computed from the results of a Coulomb excitation
experiment, indicating that this xenon isotope has a prominent triaxial ground
state. Within a different context, it was recently argued that the analysis of
particle correlations in the final states of ultra-relativistic heavy-ion
collisions performed at the Large Hadron Collider (LHC) points to a similar
structure for the adjacent isotope, $^{129}$Xe. In the present work, we report
on state-of-the-art multi-reference energy density functional calculations that
combine projection on proton and neutron number as well as angular momentum
with shape mixing for the three isotopes $^{128,129,130}$Xe using the
Skyrme-type pseudo-potential SLyMR1. Exploring the triaxial degree of freedom,
we demonstrate that the ground states of all three isotopes display a very
pronounced triaxial structure. Moreover, comparison with experimental results
shows that the calculations reproduce fairly well the low-energy excitation
spectrum of the two even-mass isotopes. By contrast, the calculation of
$^{129}$Xe reveals some deficiencies of the effective interaction.
