  In Twin Higgs models the dominant source of fine-tuning is the cancellation
of order $v^2/f^2$ required to obtain a Standard Model-like Higgs, where $v$
and $f$ are the electroweak and new physics scales, respectively. Recently
proposed Gegenbauer Goldstone models naturally realise $v^2/f^2 \ll 1$ and
hence remove this source of fine-tuning. By combining the two into
`Gegenbauer's Twin', we obtain a symmetry-based model for Higgs sector
naturalness consistent with current collider measurements without fine-tuning
of parameters. Single-Higgs coupling deviations of a few percent and trilinear
self-coupling deviations of order one are irreducible in the natural parameter
space. Thus, notably, the fingerprints of Gegenbauer's Twin could emerge first
through di-Higgs measurements at the High-Luminosity LHC.
