I have two bridges, and each one has a swivel...and it works really well for me.
I have no issues with keeping my HH and hitch lined up. I have been using them for almost a year with great results. If indeed the whole lot does twist, it is a matter of seconds or less to twist it back.
What I do like is when I have to swivel myself around and through things, a quick visual check of my HH and hitch and the whole lot has followed me, a quick twist and it's all oriented again, no twisted bridges or having to perform a circular dance in the air to untwist things.
I tried the 'fixed' swivel first, one that only twists in one plane, and then I noticed on the included paperwork that it should NOT be loaded where the length of it is against a hard point, like if you were leaning against a branch and the top and bottom parts of the swivel were sticking out free and the middle bit was taking the load!
I then switched to the DMM nexus and focus (could be wrong on the name of the focus)...anyhow, the main thing is they have shackles top and bottom that allows the swivels to 'bend'...
You can pretty much 'build' them the way you want them, fixed top or bottom, shackle top or bottom...or both.