The family of III-V element compounds (i.e., XY compounds; X = B, Al, Ga, In, or Tl; Y = N, P, As, or Sb) have been intensively investigated for several decades because of their enormous applications for many optoelectronic devices. Here, by employing first-principles calculations, the electronic structures of bulk XY haeckelite compounds are examined. It is identified that InSb (TlN and TlP) is Dirac semimetal (are strong topological insulators). The other fifteen XY compounds are semiconducting. The effect of biaxial and uniaxial tensile and compressive strains on the electronic structures are studied. These materials offer diverse topological orders. The semiconducting band gaps are mainly found between the bonding and antibonding states of the mixed X(p)-Y(p) orbitals at the top of the valence band and the bottom of the conduction bands, respectively. The topological insulating nature of the XY compounds is explained based on the degenerate p(x) + p(y) orbitals and their orbital energies relative to the p(z) orbitals near the Fermi energy. The nontrivial band topologies of TlN and TlP are confirmed by calculating the Z(2) (1;000) index, surface states, and Wilson loop calculations. The bands split into two branches by including spin-orbit interaction. The results demonstrate that haeckelite compounds are fascinating materials with broad potential applications in optoelectronics and possessing the possibility of hosting emergent physical phenomena.