Hello Rob,
No the stalks are not the same on a W116. They are completely different even down to the plugs. The only stalk which is close is a W108 250S, SE , this unit may function in a W113 but if I recall the angle or length of the arm is a little different.
Headrests from many models will work W108, W116, W123, W115, W114, W107, W109. In general these are models 1969 through 1985 era. Correct versions in early sedans and SLs up to around 1972 had the correct hole in each side, but all these later ones work just the same without the hole, (a minor difference). I will try to post some pictures of these versions asap.
Joe Alexander
Blacklick, Ohio