  We show that, for certain families $\phi_{\mathbf{s}}$ of diffeomorphisms of
high-dimensional spheres, the commutator of the Dehn twist along the
zero-section of $T^*S^n$ with the family of pullbacks $\phi^*_{\mathbf{s}}$
gives a noncontractible family of compactly-supported symplectomorphisms. In
particular, we find examples: where the Dehn twist along a parametrised
Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation;
where the symplectomorphism group is not simply-connected, and where the
symplectomorphism group does not have the homotopy-type of a finite CW-complex.
We show that these phenomena persist for Dehn twists along the standard
matching spheres of the $A_m$-Milnor fibre. The nontriviality is detected by
considering the action of symplectomorphisms on the space of parametrised
Lagrangian submanifolds. We find related examples of symplectic mapping classes
for $T^*(S^n\times S^1)$ and of an exotic symplectic structure on
$T^*(S^n\times S^1)$ standard at infinity.
