Dear Henry:
The term hyperreflexia can mean several things. Classically, it means that when any reflex is elucidated there is a non-physically response that is greater than seen in the normal population. We grade reflexes on a scale of 0-4. Zero is no reflex with re-enforcement or counter pressure, 1 is reflex with counter pressure, 2 is normal, 3 is increased but not pathological, and 4 is marked hyperactive and often associated with clonus. What you are describing is cross adduction, this is where the reflex is elicited on the left side of the body and a reflex is seen on the other side or right side of the body. This would occur is there is lack of proper inhibition. This could occur from lack of CNS input anywhere from the brain thru the spinal cord. Note that pathological hyperactive reflexes occur without cross adduction. This is also centrally driven (CNS input from brain to spinal cord).
I hope I did not confuse you. Reflex hyper-active states are from central process abnormality that may or may not show cross adduction reflexes. Such processes are things like stroke, MS, infection, abcess (abscess), spinal cord damage, tumors, etc.
Sincerely,
CCF Neuro MD