Abstract We report the discovery of three stars that, along with the prototype LP 40−365, form a distinct class of chemically peculiar runaway stars that are the survivors of thermonuclear explosions. Spectroscopy of the four confirmed LP 40−365 stars finds ONe-dominated atmospheres enriched with remarkably similar amounts of nuclear ashes of partial O- and Si-burning. Kinematic evidence is consistent with ejection from a binary supernova progenitor; at least two stars have rest-frame velocities indicating they are unbound to the Galaxy. With masses and radii ranging between 0.20 and 0.28 M$⊙$ and between 0.16 and 0.60 R$⊙$, respectively, we speculate these inflated white dwarfs are the partly burnt remnants of either peculiar Type Iax or electron-capture supernovae. Adopting supernova rates from the literature, we estimate that ∼20 LP 40−365 stars brighter than 19 mag should be detectable within 2 kpc from the Sun at the end of the Gaia mission. We suggest that as they cool, these stars will evolve in their spectroscopic appearance, and eventually become peculiar O-rich white dwarfs. Finally, we stress that the discovery of new LP 40−365 stars will be useful to further constrain their evolution, supplying key boundary conditions to the modelling of explosion mechanisms, supernova rates, and nucleosynthetic yields of peculiar thermonuclear explosions.