Axial octupole and tetrahedral correlations in ⁸⁰Zr and ⁹⁸Zr have been investigated using the generator coordinate method, applied to a basis generated by Skyrme Hartree-Fock+BCS calculations. We focus on the possible presence of states with tetrahedral symmetry and their stability with respect to octupole vibrations. We show that pairing significantly reduces the stability of the tetrahedral configuration and that a shallow mean-field tetrahedral minimum coexists with an axial octupole minimum. The contributions to the correlation energies coming from the tetrahedral degree of freedom and octupole axial deformation are discussed.