  This is the first of two companion papers in which a thorough study of the
normal form and the first integrability conditions arising from {\em
bi-conformal vector fields} is presented. These new symmetry transformations
were introduced in {\em Class. Quantum Grav.} \textbf{21}, 2153-2177 and some
of their basic properties were addressed there. In our calculations a new
affine connection ({\em bi-conformal connection}) arises quite naturally and
this connection enables us to find a local characterization of {\em conformally
separable} pseudo-Riemannian manifolds (also called double twisted products) in
terms of the vanishing of a rank three tensor $T_{abc}$. Similar local
characterizations are found for the most important particular cases such as
(double) warped products, twisted products and conformally reducible spaces
(this classification will be enlarged in the forthcoming second paper).We
speculate with potential applications of the bi-conformal connection to the
invariant characterization of other types of pseudo-Riemannian manifolds in
which the metric tensor has a part conformal to a metric of lesser rank.
