I have a galvanized chimney, no bricks, no stones, no tile liner, just
one galvanized sheet metal pipe within another, to keep the outside
cool, and an oil furnace in the basement.

Is there supposed to be a cleanout in the house? Can I/should I put
one in?

No one who has been here to clean the furnace has ever said a word
about cleaning the chimney. And I usually watched, so I know they
didn't do it.

I also have a metal chimney for the fireplace. Each chimeny has a
metal chimney cap. Because I thought wood made fireplace chimneys
dirty, I even had a chimney sweep to clean the fireplace chimney and
after he left, I was sure he cleaned that one only and didn't even ask
me if I wanted the other one cleaned. I think I would have said yes.

Does it need it? I found a brush and poles at 80% off so I bought a
set but I can't go up on the roof in the winter. Brick chimneys have
cleanouts, but before the closet under the steps was full of junk,
I've been behind the furnace and I sure don't remember seeing anything
like that. Is it possible that my galvanized chimney has no
cleanout?

Should I put one in? Would it be hard? IIRC the chimney starts about
six feet above the basement floor.

TIA