Motivated by the recent discovery of a canyon of conductance suppression in a two-level equal-spin quantum dot system [Phys. Rev. Lett. 104, 186804 (2010)], the transport through this system is studied in detail. At low bias and low temperature a strong current suppression is found around the electron-hole symmetry point independent of the couplings, in agreement with previous results. By means of a Schrieffer–Wolff transformation we are able to give an intuitive explanation to this suppression in the low-energy regime. In the general situation, numerical simulations are carried out using quantum rate equations. The simulations allow for the prediction of how the suppression is affected by the couplings, the charging energy, the position of the energy levels, the applied bias, and the temperature. We find that, away from electron-hole symmetry, the parity of the couplings is essential for the current suppression. It is also shown how broadening, interference, and a finite interaction energy cause a shift of the current minimum away from degeneracy. Finally we see how an increased population of the upper level leads to current peaks on each side of the suppression line. At sufficiently high bias we discover a coherence-induced population inversion.