  We explore a one parameter $\zeta$-deformation of the quantum-mechanical
Sine-Gordon and Double-Well potentials which we call the Double Sine-Gordon
(DSG) and the Tilted Double Well (TDW), respectively. In these systems, for
positive integer values of $\zeta$, the lowest $\zeta$ states turn out to be
exactly solvable for DSG - a feature known as Quasi-Exact-Solvability (QES) -
and solvable to all orders in perturbation theory for TDW. For DSG such states
do not show any instanton-like dependence on the coupling constant, although
the action has real saddles. On the other hand, although it has no real
saddles, the TDW admits all-orders perturbative states that are not
normalizable, and hence, requires a non-perturbative energy shift. Both of
these puzzles are solved by including complex saddles. We show that the
convergence is dictated by the quantization of the hidden topological angle.
Further, we argue that the QES systems can be linked to the exact cancellation
of real and complex non-perturbative saddles to all orders in the
semi-classical expansion. We also show that the entire resurgence structure
remains encoded in the analytic properties of the $\zeta$-deformation, even
though exactly at integer values of $\zeta$ the mechanism of resurgence is
obscured by the lack of ambiguity in both the Borel sum of the perturbation
theory as well as the non-perturbative contributions. In this way, all of the
characteristics of resurgence remains even when its role seems to vanish, much
like the lingering grin of the Cheshire Cat. We also show that the perturbative
series is Self-resurgent -a feature by which there is a one-to-one relation
between the early terms of the perturbative expansion and the late terms of the
same expansion -which is intimately connected with the Dunne-\"{U}nsal
relation. We explicitly verify that this is indeed the case.
