Abstract: I will show that, for an approximate SO(N) global symmetry spontaneously broken by the fundamental representation there is an infinite class of radiatively stable pseudo-Nambu-Goldstone (pNGB) potentials which are Gegenbauer polynomials. This observation will be used to construct new classes of pNGB Higgs models, “Gegenbauer Higgs” and “Gegenbauer’s Twin”, which are significantly less fine-tuned than previously considered pNGB Higgs scenarios. In particular, the latter requires no fine-tuning to be rendered consistent with LHC measurements and exhibits the smoking gun signal of large Higgs self-coupling modifications which may be discoverable at the HL-LHC.