This paper aims to systematically study mystic reflection groups that emerged independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372] by the authors and in the paper [Algebr. Represent. Theory 13 (2010), 127-158] by Kirkman, Kuzmanovich and Zhang. A detailed analysis of this class of groups reveals that they are in a nontrivial correspondence with the complex reflection groups G(m, p, n). We also prove that the group algebras of corresponding groups are isomorphic and classify all such groups up to isomorphism.