Just stumbled onto this . . .
Fundamentally, Bridge Assurance is a modification to RSTP wherein both sides send BPDUs to each other. Unlike 802.1d, RSTP bridges always send BPDUs on their designated ports, regardless of whether or not the bridge received a BPDU from the root. Bridge Assurance changes this so the non-designated bridge also sends a BPDU. It becomes a "hello" mechanism between the two bridges. If one side stops receiving BPDUs, it blocks on that port in case the port became unidirectional.
It's a cool feature, but if not well-understood you can cause ports to block in the wrong places. We don't use it.