They are supposed to be opposites, and let's presume they cancel each other out ...
Well, no, let's not - at least not without good reason.
Positive mass and negative mass have oppositely-signed masses. Why would that mean they'd be opposite in all other ways?
I'm no particle physicist, but negative mass seems to integrate very poorly into the system here.
Or it could just be that, not being a particle physicist, integrating negative mass into "the system" could be beyond you. It's certainly beyond me, so I'm not even going to try.
And presumably negative mass would have particles of some sort
That's an awful lot of presumptions.