  The gravitational perturbations of a rotating Kerr black hole are notoriously
complicated, even at the linear level. In 1973, Teukolsky showed that their
physical degrees of freedom are encoded in two gauge-invariant Weyl curvature
scalars that obey a separable wave equation. Determining these scalars is
sufficient for many purposes, such as the computation of energy fluxes.
However, some applications -- such as second-order perturbation theory --
require the reconstruction of metric perturbations. In principle, this problem
was solved long ago, but in practice, the solution has never been worked out
explicitly. Here, we do so by writing down the metric perturbation (in either
ingoing or outgoing radiation gauge) that corresponds to a given mode of either
Weyl scalar. Our formulas make no reference to the Hertz potential (an
intermediate quantity that plays no fundamental role) and involve only the
radial and angular Kerr modes, but not their derivatives, which can be
altogether eliminated using the Teukolsky--Starobinsky identities. We expect
these analytic results to prove useful in numerical studies and for extending
black hole perturbation theory beyond the linear regime.
