  Given a right-non-degenerate set-theoretic solution $(X,r)$ to the
Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on
$X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation
for $r$). This family includes the original solution and the classical derived
solution. All these solutions induce isomorphic actions of the braid
group/monoid on $X^n$. The structure monoids of $r$ and $r^{(k)}$ are related
by an explicit bijective $1$-cocycle-like map. We thus turn reflections into a
tool for studying YBE solutions, rather than a side object of study. In a
different direction, we study the reflection equation for non-degenerate
involutive YBE solutions, show it to be equivalent to (any of the) three
simpler relations, and deduce from the latter systematic ways of constructing
new reflections.
