oh. I love these questions. Handshake author here. Handshake provides a way for Alice and Bob to generate a large pool of one-time pre-shared keys as well as a set of one-time lookup hashes to go with these keys. Both the lookup hashes and the pre-shared keys are generated using separate KDF processes and the input bytes are destroyed upon generation. This pool of "lookup hashes" are unique to both Alice and Bob and can be used in any order. There is no further coordination of future key generation, so this is very much the "impractical One-Time Pad" pattern. In fact, there is no reason that a true XOR based OTP couldn't be implemented as a "strategy" for Handshake.
So, no ratchets, just a bunch of "dumb" one-time-use symmetric keys with a lookup table for each chat participant. The value prop here is that, for some reason, you suspect that Eve is capable of cracking assumptions of difficulty or implementations of PKI at a level in which you want a symmetric-only crypto alternative. This puts Handshake squarely in "cyber-punk vanity project".
Operationally, the biggest known risk with Handshake is device security. If Eve was able to get a hold of your device, she'd be able to decrypt all future messages related to that chat conversation (though she wouldn't be able to decrypt messages before interception, since keys used by both the sender and receiver are aggressively destroyed upon use).