Comparing orthogonal calculus and calculus with Reality

We show that there exists a suitable $C_2$-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus ``up to a shift'' in an analogous way with the recovery of real topological $K$-theory from Atiyah's $K$-theory with Reality via appropriate $C_2$-fixed points.