The authors show how this may be consistent with a structural model where trend shocks enter the cycle, or cyclic shocks enter the trend and that identification restrictions are necessary to make this structural distinction. A reduced-form unrestricted version such as that of Morley, Nelson and Zivot (2003) is compatible with either option, but cannot distinguish which is relevant. They discuss economic interpretations and implications using US real GDP data.
Trend-Cycle Decomposition: Implications from an Exact Structural Identification
WP 13-22 - A well-documented property of the Beveridge-Nelson trend-cycle decomposition is the perfect negative correlation between trend and cycle innovations.