The canonical involution in the space of connections of a $(J^2=\pm 1)$-metric manifold

A $(J^2=\pm 1)$-metric manifold has an almost complex or almost product structure $J$ and a compatible metric $g$. We show that there exists a canonical involution in the set of connections on such a manifold, which allows to define a projection over the set of connections adapted to $J$. This projection sends the Levi Civita connection onto the first canonical connection. In the almost Hermitian case, it also sends the ${\mathrm{\nabla }}^{-}$ connection onto the Chern connection, thus applying the line of metric connections defined by ${\mathrm{\nabla }}^{-}$ and the Levi Civita connections onto the line of canonical connections. Besides, it moves metric connections onto metric connections.