  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 $\nabla^{-}$
connection onto the Chern connection, thus applying the line of metric
connections defined by $\nabla ^{-}$ and the Levi Civita connections onto the
line of canonical connections. Besides, it moves metric connections onto metric
connections.
