Supplemental rotational inertia devices provide an efficient means of suppressing ground-induced vibrations over a large range of structural periods. The beneficial effects of the inerter can be further enhanced by coupling it with a clutch system that prevents it from driving the structural response and ensures that its supplemental rotational inertia is only employed to resist the motion. In this paper, we examine the behaviour of single-degree-of-freedom and multi-degree-of-freedom structures equipped with twin inerter-clutch devices subjected to strong ground-motion. The influence of the clutch stiffness, gears play, viscous damping and dry friction, on the non-linear dynamics of the system are explored first, by analysing the stable periodic solutions of a structure with inerters under harmonic-sweeps. We demonstrate that, for the range of parameters typically expected in earthquake engineering practice, the influence of dry-friction and clutch damping are limited, although the clutch stiffness and gear play may need to be accounted for when large inertances or defective clutches are considered. Based on these findings, we propose a simplified numerical modelling strategy suitable for implementation in conventional Finite Element simulations. Small scale experiments on bare elastic structures as well as structures equipped with 3D-printed inerter and inerter-clutch twins are presented and employed for concept demonstration and for the validation of the numerical model proposed. Finally, a series of studies on detailed numerical models of multi-storey steel frames under idealized and real pulse-like ground-motions are used to demonstrate the vibration absorbing capabilities brought about by the twin inerter-clutch system and to highlight practical aspects related to their structural implementation.