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The bifree double Burnside ring $B^\Delta(G,G)$ of a finite group $G$ has anatural anti-involution. We study the group $B^\Delta_\circ(G,G)$ of orthogonalunits in $B^\Delta(G,G)$. It is shown that this group is always finite andcontains a subgroup isomorphic to $B(G)^\times\rtimes \Out(G)$, where$B(G)^\times$ denotes the unit group of the Burnside ring of $G$ and $\Out(G)$denotes the outer automorphism group of $G$. Moreover it is shown that if $G$is nilpotent then $B^\Delta_\circ(G,G)\cong B(G)^\times\rtimes \Out(G)$. Theresults can be interpreted as positive answers to questions on equivalences of$p$-blocks of group algebras in the case that the block is the group algebra ofa $p$-group.

Orthogonal units of the bifree double Burnside ring