  We show that braided Cherednik algebras introduced by the first two authors
are cocycle twists of rational Cherednik algebras of the imprimitive complex
reflection groups $G(m,p,n)$, when $m$ is even. This gives a new construction
of mystic reflection groups which have Artin-Schelter regular rings of quantum
polynomial invariants. As an application of this result, we show that a braided
Cherednik algebra has a finite-dimensional representation if and only if its
rational counterpart has one.
