  We find necessary and sufficient conditions for the bi-Legendrian connection
$\nabla$ associated to a bi-Legendrian structure $(\cal F,\cal G)$ on a contact
metric manifold $(M,\phi,\xi,\eta,g)$ being a metric connection and then we
give conditions ensuring that $\nabla$ coincides with the (generalized)
Tanaka-Webster connection of $(M,\phi,\xi,\eta,g)$. Using these results, we
give some interpretations of the Tanaka-Webster connection and we study the
interactions between the Tanaka-Webster, the bi-Legendrian and the Levi Civita
connection in a Sasakian manifold.
