In this paper, we construct a version of orthogonal calculus for functors from \(C_2\) -representations to \(C_2\) -spaces, where \(C_2\) is the cyclic group of order 2. For example, the functor \(BO(-)\) , that sends a \(C_2\) -representation to the classifying space of its orthogonal group, which has a \(C_2\) -action induced by the action on the \(C_2\) -representation. We obtain a bigraded sequence of approximations to such a functor, and via a zig-zag of Quillen equivalences, we prove that the homotopy fibres of maps between approximations are fully determined by orthogonal spectra with a genuine action of \(C_2\) and a naive action of the orthogonal group \(O(p,q):=O(\mathbb {R}^{p+q\delta })\) .
