When people approve documents, it just doesn't seem right to say that they sign "off on" them. By itself, "off on" sounds like an oxymoron, but if you use it in context, it slips by unnoticed."

I must assume it is correct usage, but my attempts to verify it proved to be unfruitful. There are lots of ways to use the verb "signed." You can sign on, off, in, for, into, out of, up, with, etc. These are called verbal phrases. I guess if you can sign off and can sign on, you should be able to sign 'off on' without anyone, other than me, questioning you.

It is not an "uncommon expression to "go off on someone," or to "go off on a tangent." Golfers tee off on hole #1..., addicts get off on drugs..., wet paint will rub off on your clothes..., and you can start off on the wrong foot, as I might have done. However, I cannot find any usage of "on off." It seems like, if "off on" is so popular, "on off" should at least appear occasionally. I guess a person can relax "on off" days, but I think it is supposed to be written "off-days,"... or maybe not.

"Off on" still doesn't seem right until you use it in a sentence, but then for some strange reason, it sounds fine. Perhaps to sign "off on" something is proper usage, or perhaps we have just adopted a slang expression, as we have with so many others. But does anyone really care? English sucks. I'll sign off on this now and go off on a more productive project.