Is C25519/Ed25519 "twist secure"? This recent new curve mentions something that's new to me: twist security.

http://safecurves.cr.yp.to/bada55.html

Are the existing C25519/Ed25519 curves secure against this form of attack?
Yes.

There are several different kinds of twist attacks. Curve25519 is safe against all the ones SafeCurves looks at. Specifically, the cost of a combined attack is $2^{124.3}$, which is considered safe.