  We consider a framework where the Standard Model is augmented by a second
SU(2) scalar doublet and by a real scalar singlet that, protected by a Z2
symmetry, provides a particle Dark Matter candidate. We show that this setup
allows for doubly blind spots at both collider searches for anomalies in the
Higgs invisible decay width, and at direct Dark Matter detection. The blind
spots originate from cancellations between interfering diagrams featuring
different neutral scalar exchanges, and from cancellations driven by the
two-Higgs doublet structure in the vertex coupling the singlet state with the
Standard-Model-like Higgs. We demonstrate that the blind spots arise in a wide
and generic array of realizations for the two-Higgs doublet model, including
scenarios with a non-trivial flavor structure. We provide analytical
formul{\ae} that describe the location of the blind spots in the theory
parameter space, and we discuss the resulting phenomenology.
