It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z 2 -orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Main examples are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra. Remark 1.10. The reflection equation algebra O q (G) is an (equivariant) quantization of the Semenov-Tian-Shansky Poisson bracket on O(G) [Mud06]. This algebra goes by many names: Majid's braided dual of U q (g) [Maj95], the quantum-loop algebra [AS96] and is isomorphic to the locally finite part of U q (g) via the Rosso form [KS09, Proposition 2.8].6