  Recently the authors initiated an $\imath$Hall algebra approach to
(universal) $\imath$quantum groups arising from quantum symmetric pairs. In
this paper we construct and study BGP type reflection functors which lead to
isomorphisms of the $\imath$Hall algebras associated to acyclic
$\imath$quivers. For Dynkin quivers, these symmetries on $\imath$Hall algebras
induce automorphisms of universal $\imath$quantum groups, which are shown to
satisfy the braid group relations associated to the restricted Weyl group of a
symmetric pair; conjecturally these continue to hold for acyclic
quivers/Kac-Moody setting. This leads to a conceptual construction of $q$-root
vectors and PBW bases for (universal) quasi-split $\imath$quantum groups of ADE
type.
