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Two definitions for the rectfiability of hypersurfaces in Heisenberg groups$\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces,and the other on Lipschitz images of subsets of codimension-$1$ verticalsubgroups. The equivalence between these notions remains an open problem.Recent partial results are due to Cole-Pauls, Bigolin-Vittone, and Antonelli-LeDonne.  This paper makes progress in one direction: the metric Lipschitzrectifiability of $\mathbb{H}$-regular surfaces. We prove that$\mathbb{H}$-regular surfaces in $\mathbb{H}^{n}$ with $\alpha$-H\"oldercontinuous horizontal normal, $\alpha > 0$, are metric bilipschitz rectifiable.This improves on the work by Antonelli-Le Donne, where the same conclusion wasobtained for $C^{\infty}$-surfaces.  In $\mathbb{H}^{1}$, we prove a slightly stronger result: everycodimension-$1$ intrinsic Lipschitz graph with an $\epsilon$ of extraregularity in the vertical direction is metric bilipschitz rectifiable. All theproofs in the paper are based on a new general criterion for findingbilipschitz maps between "big pieces" of metric spaces.

Metric rectifiability of $\mathbb{H}$-regular surfaces with H\"older continuous horizontal normal