  We study several linear connections (the first canonical, the Chern, the well
adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those
with totally skew-symmetric torsion) which can be defined on the four geometric
types of $(J^2=\pm1)$-metric manifolds. We characterize when such a connection
is adapted to the structure, and obtain a lot of results about coincidence
among connections. We prove that the first canonical and the well adapted
connections define a one-parameter family of adapted connections, named
canonical connections, thus extending to almost Norden and almost product
Riemannian manifolds the families introduced in almost Hermitian and almost
para-Hermitian manifolds. We also prove that every connection studied in this
paper is a canonical connection, when it exists and it is an adapted
connection.
