We introduce a natural extension of the usual covariance property for the reflection equation. To solve one reflection equation we propose to use a family of reflection equations with various Drinfeld twists of the initial R-matrix. This allows us to produce non-trivial representations of the reflection equation algebra in a systematic way. As an example, we consider a twist of the Lie algebra sl(2) related to some integrable tops and to the Toda lattices associated with the D_n root system.