Reflexive homology and involutive Hochschild homology as equivariant Loday constructions

We prove that for commutative rings whose underlying abelian group is flat and in which 2 is invertible, the homotopy groups at the trivial orbit of the equivariant Loday construction of the one-point compactification of the sign-representation are isomorphic to reflexive homology as studied by Graves and to involutive Hochschild homology defined by Fern\`andez-al\`encia and Giansiracusa. We also show a relative version of these results for commutative k-algebras R with involution, whenever 2 is invertible in R and R is flat as a k-module.