  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/2Z-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.
