  We initiate a general approach to the relative braid group symmetries on
(universal) $\imath$quantum groups, arising from quantum symmetric pairs of
arbitrary finite types, and their modules. Our approach is built on new
intertwining properties of quasi $K$-matrices which we develop and braid group
symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new
symmetries on $\imath$quantum groups are obtained. We establish a number of
fundamental properties for these symmetries on $\imath$quantum groups,
strikingly parallel to their well-known quantum group counterparts. We apply
these symmetries to fully establish rank one factorizations of quasi
$K$-matrices, and this factorization property in turn helps to show that the
new symmetries satisfy relative braid relations. As a consequence, conjectures
of Kolb-Pellegrini and Dobson-Kolb are settled affirmatively. Finally, the
above approach allows us to construct compatible relative braid group actions
on modules over quantum groups for the first time.
