Final answer:
The chordae tendineae are connective tissue strands that attach the heart's valve cusps to the papillary muscles, preventing the valves from inverting into the atria during ventricular contraction.
Step-by-step explanation:
The structure that is attached to the valvular cusps of the heart and prevents the valves from swinging back into the atria is the chordae tendineae. These are strong strands of connective tissue that connect the valve cusps to the papillary muscles in the ventricles.
When the ventricles contract, the papillary muscles also contract, creating tension on the chordae tendineae, which in turn prevents the valve flaps from being pushed back into the atria and thus avoids the backflow of blood, or regurgitation, into the atria during ventricular contraction.