  Starting from the recently-discovered $\textrm{T}\bar{\textrm{T}}$-perturbed
Lagrangians, we prove that the deformed solutions to the classical EoMs for
bosonic field theories are equivalent to the unperturbed ones but for a
specific field-dependent local change of coordinates. This surprising geometric
outcome is fully consistent with the identification of
$\textrm{T}\bar{\textrm{T}}$-deformed 2D quantum field theories as topological
JT gravity coupled to generic matter fields. Although our conclusion is valid
for generic interacting potentials, it first emerged from a detailed study of
the sine-Gordon model and in particular from the fact that solitonic
pseudo-spherical surfaces embedded in $\mathbb R^3$ are left invariant by the
deformation. Analytic and numerical results concerning the perturbation of
specific sine-Gordon soliton solutions are presented.
