I'm going to re-use the yeast cake from a low-gravity batch to ferment a high-gravity batch.

• Can I just put the wort directly on it? Does it then need to be agitated?
• Is it worth the risk of contamination to take the whole yeast cake out, wash it, strain it, and actually re-pitch it?
• Is there a term for this complete re-use of a yeast cake?

UPDATE

Pouring down on the yeast cake of the previous batch is something I do regularly and with great effect. I don't remove anything from the carboy and I only step up in gravity. This might be helped by the fact that I only use leaf hops and they do a great job latching onto break material. I also use a boil screen, so there's very little trub anyway.

Yes, you can put the wort directly in. No, it doesn't need to be agitated, but a bit of shaking for aeration doesn't hurt.

I can't speak to risk of contamination with removing and repitching slurry, but I can say that the risk of contamination with pouring down is less.

FURTHER UPDATE

I still do this regularly with good results. I wanted to do some testing on how much ester production is inhibited by this degree of overpitching, so I tried making a light saison, with Wyeast French Saison (3711). (It's amazing yeast, by the way.) It was effectively Northern Brewer's Petite Saison d'Ete recipe, but at a 1.036 gravity. I made that batch, kegged it, then poured down a 1.080 biere de garde onto the cake. Fermentation was vigorous and healthy. The beer is missing nothing in the ester/character department. This beer had as much character and interest as anything I've made, even with being massively overpitched by homebrew standards. If anything it's less "clean" than I wanted.

Long story short: you can make very good beer like this, especially big beers.