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We introduce equivariant factorization homology, extending the axiomatic framework of Ayala-Francis to encompass multiplicative invariants of manifolds equipped with finite group actions. Examples of such equivariant factorization homology theories include Bredon equivariant homology and (twisted versions of) Hochschild homology. Our main result is that equivariant factorization homology satisfies an equivariant version of ⊗-excision, and is uniquely characterised by this property. We also discuss applications to representation theory, such as constructions of categorical braid group actions. 7Notation. We denote the connected components of D I = [Γ:I] R n by R n gI for gI ∈ Γ/I. 6 Where ρ is the 6-dimensional real representation underlying the 3-dimensional complex representation. 9

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

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