In a sequel to [2] the authors study algebraic curves in |R2 which are mapped into themselves by the reflection-inversionS(y):=−||y||−2y(y(yɛ|R2{(0,0)}), a birational transformation of the plane. It is shown that every curve with this property is, essentially, the combined image of two homeomorphic models of a real projective algebraic curve in the standard model ofp2(|R). Curves of low degree possessing this property are then identified.