We study the stochastic six-vertex model in half-space with generic integrable boundary weights, and define two families of multivariate rational symmetric functions. Using commutation relations between double-row operators, we prove a skew Cauchy identity of these functions. In a certain degeneration of the right-hand side of the Cauchy identity we obtain the partition function of the six-vertex model in a half-quadrant, and give a Pfaffian formula for this quantity. The Pfaffian is a direct generalization of a formula obtained by Kuperberg in his work on symmetry classes of alternating-sign matrices. One of our families of symmetric functions admits an integral (sum over residues) formula, and we use this to conjecture an orthogonality property of the dual family. We conclude by studying the reduction of our integral formula to transition probabilities of the (initially empty) asymmetric simple exclusion process on the half-line.