Starting from the Izergin-Korepin 19-vertex model in the quadrant, we introduce two families of rational multivariate functions $F_S$ and $G_S$; these are in direct analogy with functions introduced by Borodin in the context of the higher-spin 6-vertex model in the quadrant. We prove that $F_S(x_1,\dots,x_N;z)$ and $G_S(y_1,\dots,y_M;z)$ are symmetric functions in their alphabets $(x_1,\dots,x_N)$ and $(y_1,\dots,y_M)$, and pair together to yield a Cauchy identity. Both properties are consequences of the Yang-Baxter equation of the model. We show that, in an appropriate limit of the spectral parameters $z$, $F_S$ tends to a stable symmetric function denoted $H_S$. This leads to a simplified version of the Cauchy identity with a fully factorized kernel, and suggests self-duality of the functions $H_S$. We obtain a symmetrization formula for the function $F_S(x_1,\dots,x_N;z)$, which exhibits its symmetry in $(x_1,\dots,x_N)$. In contrast to the 6-vertex model, where $F^{6{\rm V}}_S(x_1,\dots,x_N;z)$ is cast as a sum over the symmetric group $\mathfrak{S}_N$, the symmetrization formula in the 19-vertex model is over a larger set of objects that we define; we call these objects 2-permutations. As a byproduct of the proof of our symmetrization formula, we obtain explicit formulas for the monodromy matrix elements of the 19-vertex model in a basis that renders them totally spatially symmetric.