  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.
