  We consider Higher-Order Scalar-Tensor theories which appear degenerate when
restricted to the unitary gauge but are not degenerate in an arbitrary gauge.
We dub them U-degenerate theories. We provide a full classification of theories
that are either DHOST or U-degenerate and that are quadratic in second
derivatives of the scalar field, and discuss its extension to cubic and higher
order theories. Working with a simple example of U-degenerate theory, we find
that, for configurations in which the scalar field gradient is time-like, the
apparent extra mode in such a theory can be understood as a generalized
instantaneous, or "shadowy" mode, which does not propagate. Appropriate
boundary conditions, required by the elliptic nature of part of the equations
of motion, lead to the elimination of the apparent instability associated with
this extra mode.
