I rolled over an HSA from one provider to another, in a direct trustee-to-trustee transfer. I had no other withdrawls from the rolled-over HSA, and the HSA provider did not send me a 1099-SA (nor is one available on their website). If my understanding is correct, this means I do not have to file anything for the rollover, since it was trustee-to-trustee. Is that correct?
you are correct nothing to file for the direct rollover. no 1099-SA will be issued.
To be clear, this is a trustee-to-trustee transfer, not a rollover. A trustee-to-trustee transfer of an HSA is not reportable. (If you instead received a distribution paid to you and within 60 days deposited this money into another HSA, that would be a reportable distribution and rollover.)
Is that specific to the context of HSAs? I've done 401(k) direct transfer rollovers to an IRA, but I still got 1099-R's for them.
That's specific to HSA to HSA, traditional IRA to traditional IRA and Roth IRA to Roth IRA (tracing back to IRS Revenue Ruling 78-406 which defines a trustee-to-trustee transfer, a rule established to allow movement of these types of accounts from one custodian to another without regard to the one-rollover-per-12-months limitation by it being neither a distribution nor a rollover). Movement of funds to or from a 401(k) is always reportable (except for the case where one company acquires another resulting in the merger of two 401(k) plans).