  Given a finite group $G$, its double Burnside ring $B(G,G)$, has a natural
duality operation that arises from considering opposite $(G,G)$-bisets. In this
article, we systematically study the subgroup of units of $B(G,G)$, where
elements are inverse to their dual, so called orthogonal units. We show the
existence of an inflation map that embeds the group of orthogonal units of
$B(G/N,G/N)$ into the group of orthogonal units of $B(G,G)$, when $N$ is a
normal subgroup of $G$, and study some properties and consequences. In
particular, we use these maps to determine the orthogonal units of $B(G,G)$,
when $G$ is a cyclic $p$-group, and $p$ is an odd prime.
