We theoretically demonstrate how competition between band inversion and spin-orbit coupling (SOC) results in nontrivial evolution of band topology, taking antiperovskite Ba3SnO 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 A3EO 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.Singularities in band structures give rise to singular and attracting responses of materials [1][2][3][4][5]. Therefore, significant amount of efforts have been paid on finding what kinds of singularities are possible in principle, and on proposing materials to realize the singularities, both in theory and experiments [6][7][8]. The "standard" singularity is Dirac/Weyl cone [9-13], which is characterized by a linear, or conical dispersion around an isolated gapclosing point in the Brillouin zone. Already a number of materials are confirmed to possess Dirac/Weyl cones, ranging from graphene [14] (two-dimensional) to Cd 3 As 2 [15], Na 3 Bi [16], and TaAs [17] (three-dimensional), and so on. However, the linear dispersion around the isolated gap-closing point is not the only possible band singularity. The gap-closing point can form a line (typically a loop) in the Brillouin zone rather than an isolated point [18][19][20]. Also, a dispersion around a gap-closing point can be quadratic rather than linear [21]. Very recently, singularities involving three bands are also discussed [22,23].Quite often, the three ingredients, i.e., band inversion, spin-orbit coupling (SOC), and symmetry, play key roles in generating a singular band structure. Roughly speaking, the band inversion means an overlap between two bands with different origin, typically originating from different kinds of orbitals, say s-and p-orbitals, or p-and d-orbitals. Naively, the overlapped bands repel with each other (band anticrossing). However the gap opening due to the band repulsion is sometimes prohibited by some symmetry, leading to singular gap closing points. The SOC, on the other hand, determines which kinds of the band repulsion is allowed, in other words, what kinds of gap-closing points can appear. terial Ca 3 PbO is predicted to have three-dimensional Dirac cones exactly at the Fermi energy, and importantly, all of the three ingredients, the band overlap, SOC, and the symmetry, are relevant in the formation of Dirac cones. (Strictly speaking, there is a tiny mass gap. If we make an emphasis on the gapful nature, the system is a topological crystalline insulator ...