We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A,μ, u) in a monoidal category, as a morphism T : A⊗ A → A ⊗ A satisfying a list of axioms ensuring that (A,μ ◦ T, u) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted ten...