  This paper studies the inverse Steklov spectral problem for curvilinear
polygons. For generic curvilinear polygons with angles less than $\pi$, we
prove that the asymptotics of Steklov eigenvalues obtained in arXiv:1908.06455
determines, in a constructive manner, the number of vertices and the properly
ordered sequence of side lengths, as well as the angles up to a certain
equivalence relation. We also present counterexamples to this statement if the
generic assumptions fail. In particular, we show that there exist non-isometric
triangles with asymptotically close Steklov spectra. Among other techniques, we
use a version of the Hadamard--Weierstrass factorisation theorem, allowing us
to reconstruct a trigonometric function from the asymptotics of its roots.
