Is there a knot that can be used to hitch a rope to a pole without access to any of the rope's ends OR the pole's ends? I know of hitches that work in the one case or the other but not both.
For what purposes? For 'general' purpose, you should be able to get by just fine by making a bight in your line, then tying any knot you'd normally use in your situation using the bight as your line. This is known as tying a knot 'in the bight'.
For example, the classic bowline could be used with the rope doubled up (aka; a double bowline), as can two half hitches or the anchor bend. A rolling hitch may work if your knot needs to resist sliding along the pole.
Just be sure to leave plenty of slack on the free end to tie an overhand backup knot, or clip a carabiner or something to the free loop as backup to keep the knot from pulling out.
Image Source: Mountaineering: The Freedom of the Hills, 8th Edition
The Blake's Hitch tied with with a bight should work okay as well.
It is a tree climbing/friction/sliding knot very similar to a prussik or klemheist but without needing a loop for its structure. Tying it with a bight end and either 3 or 4 wraps (of a bight will create either 6 or 8 actual wraps) will allow the rope to grip the pole. Most people will put some sort of stopping knot in the end of the bight end but I've personally never had it come loose while tree climbing.