This work investigates the suppression of individual harmonics, simultaneously affecting specific even and odd orders in the high-harmonic spectra generated by strongly tailored, two-colour, multi-cycle laser pulses in neon. The resulting spectra are systematically studied as a function of the electric-field shape in a symmetry-broken (ω-2ω) and symmetry-preserved (ω-3ω) configuration. The peak suppression is reproduced by macroscopic strong-field approximation calculations and is found to be unique to symmetry-broken fields (ω-2ω). Additionally, semi-classical calculations further corroborate the observation and reveal their underlying mechanism, where a nontrivial spectral interference between subsequent asymmetric half-cycles is found to be responsible for the suppression.