  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--P\'ech\'e 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.
