We introduce the notion of a (quasi-)reflection algebra over a quasitriangular (quasi-)bialgebra. In earlier joint work with P. Etingof, we constructed a dynamical pseudotwist for each triple (g,t,sigma), where (g,t) is a Lie algebra with an invariant quadratic element and sigma is an automorphism of finite order N. We show that this pseudotwist gives rise to examples of quasi-reflection algebras. We define a scheme of universal pseudotwists with distribution relations; it has a natural morphism to the scheme of associators. We construct rational points in this scheme. We also show that it has the structure of a torsor over a group GTMD(N,k), which maps to Drinfeld's group GT(k). We study the associated graded Lie algebra with (Z/NZ)^\times-action grtmd_1(N,k), and discuss relations with Gal(barQ/Q).