  We theoretically demonstrate how competition between band inversion and
spin-orbit coupling (SOC) results in nontrivial evolution of band topology,
taking antiperovskite Ba$_3$SnO as a prototype material. A key observation is
that when the band inversion dominates over SOC, there appear "twin" Dirac
cones in the band structure. Due to the twin Dirac cones, the band shows highly
peculiar structure in which the upper cone of one of the twin continuously
transforms to the lower cone of the other. Interestingly, the relative size of
the band inversion and SOC is controlled in this series of antiperovskite
A$_3$EO by substitution of A (Ca, Sr, Ba) and/or E (Sn, Pb) atoms. Analysis of
an effective model shows that the emergence of twin Dirac cones is general,
which makes our argument a promising starting point for finding a singular band
structure induced by the competing band inversion and SOC.
