  For almost contact metric or almost paracontact metric manifolds there is
natural notion of $\eta$-normality. Manifold is called $\eta$-normal if is
normal along kernel distribution of characteristic form. In the paper it is
proved that $\eta$-normal manifolds are in one-one correspondence with
Cauchy-Riemann almost contact metric manifolds or para Cauchy-Riemann in case
of almost paracontact metric manifolds. There is provided characterization of
$\eta$-normal manifolds in terms of Levi-Civita covariant derivative of
structure tensor. It is established existence a Tanaka-like connection on
$\eta$-normal manifold with autoparallel Reeb vector field. In particular case
contact metric CR-manifold it is usual Tanaka connection. Similar results are
obtained for almost paracontact metric manifolds. For manifold with closed
fundamental form we shall state uniqueness of this connection. In the last part
is studied bi-Legendrian structure of almost paracontact metric manifold with
contact characteristic form. It is established that such manifold is
bi-Legendrian flat if and only if is normal. There are characterized semi-flat
bi-Legendrian manifolds.
