Well the HBK2 has a 120° swash config ... right? Which is called CCPM. Let's start there.
On PG 89 of the manual ... it tells you how to turn CCPM ON. The 120° swash config appears to be the only CCPM swash config that the DX6 supports so I guess you lucked out there.

With CCPM, if all the servos are working correctly with relative to each other but are in the wrong direction relative to stick movement ... you must reverse the FUNCTION (Pitch in your case) not the (pitch) CHANNEL.

This is where the instructions get a little sketchy.
Step 6 sets the value for each channel. Actually you are setting the mix for each FUNCTION not channel. This is obviously turned on or your pitch function wouldn't work right. So if the pitch FUNCTION is working right but the direction relative to stick movement is wrong ... the FUNCTION needs reversed.

Anyway ... step 7 says: " 7. Press the SCROLL button once to access the servo reversing function." I believe this is how you reverse your FUNCTION (they call it "servo"). There is a different menu/screen to reverse a channel (servo) ... PG72
The steps on PG89 are how the CCPM is activated (turned ON) and how it is set up.

So ... PG72 is how to reverse (the swash channels) CHANNELS and PG89 is how to reverse (the swash) FUNCTIONs. In your description, it appears you need to reverse the pitch FUNCTION.