At present the three major unknowns in neutrino oscillation parameters are the mass hierarchy, the octant of $theta_{23}$ and the CP phase $delta_{CP}$. It is well known that the presence of hierarchy$-delta_{CP}$ and octant degeneracies affects the unambiguous determination of these parameters. In this paper we show that a comprehensive way to study the remaining parameter degeneracies is in the form of a generalized hierarchy-$theta_{23}$ - $delta_{CP}$ degeneracy. We show that the wrong-hierarchy and/or wrong-octant solutions can be further classified into eight different solutions depending on whether they occur with the wrong or right value of $delta_{CP}$. These eight solutions are different from the original eightfold degenerate solutions and can exist, in principle, even if $theta_{13}$ is known. These multiple solutions, apart from affecting the determination of the true hierarchy and octant, also affect the accurate estimation of $delta_{CP}$. We identify which of these eight different degenerate solutions can occur in the test ($theta_{23} - delta_{CP}$) parameter space, taking the long-baseline experiment NO$ u$A running in the neutrino mode as an example. The inclusion of the NO$ u$A antineutrino run removes the wrong-octant solutions appearing with both right and wrong hierarchy. Adding T2K data to this resolves the wrong hierarchy -- right octant solutions to a large extent. The remaining wrong hierarchy solutions can be removed by combining NO$ u$A + T2K with atmospheric neutrino data. We demonstrate this using ICAL@INO as the prototype atmospheric neutrino detector. We find that the degeneracies can be resolved at the $2sigma$ level by the combined data set, for the true parameter space considered in the study.
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