  Building on arXiv:1902.03807, this paper develops a unifying study on the
boundedness properties of several representative classes of hybrid operators,
i.e. operators that enjoy both zero and non-zero curvature features.
Specifically, via the LGC-method, we provide suitable $L^p$ bounds for three
classes of operators: (1) Carleson-type operators, (2) Hilbert transform along
variable curves, and, taking the center stage, (3) Bilinear Hilbert transform
and bilinear maximal operators along curves. All these classes of operators
will be studied in the context of hybrid curves with no quadratic resonances.
  The above study is interposed between two naturally derived topics:
  i) A prologue providing a first rigorous account on how the presence/absence
of a higher order modulation invariance property interacts with and determines
the nature of the method employed for treating operators with such a property.
  ii) An epilogue revealing how several key ingredients within our present
study can blend and inspire a short, intuitive new proof of the smoothing
inequality that plays the central role in the analysis of the curved version of
the triangular Hilbert transform treated in arXiv:2008.10140.
