Never, ever, never roto-till a horseradish plant. Even hair roots as
thick as a pencil lead will root and sprout. I dug down four feet and
had to give up. They grow to hell and take the water from the lost
souls...

They are not "invasive" as long as you do NOT make chunks of the
roots. As long as you restrict yourself to making a single cut when
harvesting on a mature specimen (dig down and remove all hair roots when
harvesting a segment), it should continue to grow true.

And they will not die. Ever. They do not feel pity, or remorse....

you will never be rid of them. Horseradish produce long tap roots - it is impossible to dig the root without breaking and every broken chunk will be the start of a new plant.