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

$\eta$-Normality, CR-structures, para-CR structures on almost contact metric and almost paracontact metric manifolds