We formulate and establish symmetries of certain integrable half space models, analogous to recent results on symmetries for models in a full space. Our starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure which may be of independent interest. As an application, we establish a distributional identity between the absorption time in a type $B$ analogue of the oriented swap process and last passage times in a half space, establishing the Baik--Ben Arous--Péché phase transition for the absorption time. The proof uses Hecke algebras and integrability of the six vertex model through the Yang--Baxter and reflection equations.