He didn't; he climbed DOWN Satan initially to reach the grotto, then climbed up the cliff to reach Purgatory.
Partway down, tho, gravity (or perspective.. hard to say which) reverses.
To quote the book:
The ice ended short of Lucifer himself. There was three feet of empty space all around the enormous waist.
And no navel. I couldn't have missed it. It would have been big enough to hide a battleship.
"You must climb down," said Benito.
I looked into the gap. "After you."
We descended, getting heavier. There was a point where I must have weighed tons, and all of it pushing inward toward my navel. No quantum black hole crushed and swallowed me. I hadn't really expected one.
Benito worked his way around until his feet pointed at me, and kept climbing. I followed his example.
Now we climbed up.
The climb seemed endless, but it ended, not in ice but in an echoing grotto of gray rock, dimly lit.
The Devil's hooves still loomed over us, big enough to stamp a city flat.
We lay on our backs on the smooth rock, panting. Somewhere a running stream made a bright, happy gurgling sound. The dim light came from a single bright pinpoint source overhead. The rock curved inward over our heads, but it never closed. It stretched away like the neck of an inverted funnel, straight up for an unguessable distance.
But Benito was on his feet. "Onward!" he cried, and began to climb. The handholds were not difficult, and he moved like a spider monkey, or like a fat man who no longer weighs anything at all.He looked down from the inward-tilting gray slope of grotto roof. "A four-thousand-mile climb, if Dante was half right!" he bellowed cheerfully. "Are you coming?"
Some clear strangeness going on with Up & Down and so forth... But he climbed down thru the ice, just as Dante described.