Energy landscapes of superheavy even-even nuclei with 116 ≤ Z ≤ 126 and 176 ≤ N ≤ 184 are studied within the macroscopic-microscopic method over a 12 dimensional manifold of shapes, including both axial- and mass-asymmetry. In addition to the spherical, prolate and oblate minima we find also oblate-octupole minima, with Y33 symmetry along the oblateness axis, that compete to become ground states. All fission saddle points are triaxial and by 0.5-2.5 MeV lower than the axial ones. The Woods-Saxon model predicts the abrupt lowering of barriers near Z = 126, in a stark contrast to the mostly used self-consistent effective interactions.