Try Unison: http://www.cis.upenn.edu/~bcpierce/unison/
Syntax:
unison dirA/ dirB/
Unison asks what to do when files are different, but you can automate the process by using the following which accepts default (nonconflicting) options:
unison -auto dirA/ dirB/
unison -batch dirA/ dirB/ asks no questions at all, and writes to output how many files were ignored (because they conflicted).
Note: I am no longer using Unison (I use NextCloud, which doesn’t address the original use case). However, note that rsync is not designed for bidirectional sync, while unison is. unison may have its bugs (as any other piece of software) and its wrinkles. I am surprised it seems to be actively maintained now (last time I looked I think I thought it looked dead), but I’m not sure what’s the state nowadays. I haven’t had the need to have a two-way file synchronizer, so there may be better options, though.