  We consider the space $\Lambda M:=H^1(S^1,M)$ of loops of Sobolev class $H^1$
of a compact smooth manifold $M$, the so-called free loop space of $M$. We take
quotients $\Lambda M/G$ where $G$ is a finite subgroup of $O(2)$ acting by
linear reparametrization of $S^1$. We use the existence of transfer maps
$tr:H_*(\Lambda M/G)\rightarrow H_*(\Lambda M)$ to define a homology product on
$\Lambda M/G$ via the Chas-Sullivan loop product. We call this product $P_G$
the transfer product. The involution $\vartheta:\Lambda M\rightarrow \Lambda M$
which reverses orientation, $ \vartheta\big(\gamma(t)\big):=\gamma(1-t)$, is of
particular interest to us. We compute $H_*(\Lambda S^n/\vartheta;\mathbb{Q})$,
$n>2$, and the product $P_\vartheta:H_i(\Lambda S^n/\vartheta;\mathbb{Q})\times
H_j(\Lambda S^n/\vartheta;\mathbb{Q})\rightarrow H_{i+j-n}(\Lambda
S^n/\vartheta;\mathbb{Q})$ associated to orientation reversal. Rationally
$P_\vartheta$ can be realized "geometrically" using the concatenation of
equivalence classes of loops. There is a qualitative difference between the
homology of $\Lambda S^n/\vartheta$ and the homology of $\Lambda S^n/G$ when
$G\subset S^1\subset O(2)$ does not "contain" the orientation reversal. This
might be interesting with respect to possible differences in the number of
closed geodesic between non-reversible and reversible Finsler metrics on $S^n$,
the latter might always be infinite.
