I have all three, with Orient being the latest to join my Japanese club. I am very impressed with the build and timing accuracy of the 2 Orient 21 jewel mechanicals - I'll agree that you aren't going to see one as often as Seiko or Citizen - that's part of the charm - for now!
As far as education of the brands, Seiko and Citizen are arch rivals while Orient is partially owned by Seiko. Orient does not yet actively market their product in North America, they do well in Eastern Europe, Middle East, South America, Africa, Middle East, Russia and their home market of Japan. FWIR, Orient is more prestigious than Seiko in Japan, they send reminder notices to their individual Japanese customers to have their timepieces serviced at the appropriate time interval!
Citizen is most well known for their Eco Drive watches, these far outsell the Seiko Kinetics. Citizen makes mechanical watch movements under their division Miyota, these are used in some Citizen branded models ( divers typically) and also supplied to other makers like Invicta, Adye Kaye, Croton, etc. Robust mechanicals, accurate quartz, and eco friendly eco drive.
Seiko is ... well you should know enough about them if you lurk around this place! Excellent value in mechanicals, quartz, high tech Spring Drive. This is the penultimate vertically integrated company, they make everything that goes into their watch, the glass, the movement, the battery, the jewels, dials, screws. I'm safe to predict that they part own the stainless steel mills too!
I compared my Orient mechanical movement(s) to that of the Seiko 7s26, and you can see that Orient has a version of the Seiko 'magic finger' winding system. In use the Orient has a far noisier winding rotor compared to the Seiko, but this is not something you feel on the wrist. The Miyota mechanical is quirky compared to the Seiko or Orient, the Miyota rotor only winds in 1 direction, the seconds hand is not direct drive so it can exhibit a 'stutter' when the watch is tapped hard - it doesn't affect accuracy at all - it only looks stupid to see the seconds hand stop momentarily and then resume its ticking. You can manually wind a Miyota, not Seiko or Orient. All 3 lack a hacking feature, but with the Orient and Seiko, slight reverse torque on the winding stem when setting the watch will cause it to stop and / or run backwards - handy to synchronise to a time signal and apparently causes no damage.
Hope this helps - I love them all, but I find the Seiko 7s26 to be the least accurate and least responsive to manual regulation - others will disagree with me, but you asked!
Thanks for the great tutorial Mtech. I am always biased towards Seiko (since my first real watch when I was growing up in the 60's was a Seiko 5) and actually own 2 Seikos right now (1 quartz dress and 1 automatic SKX031) also have an automatic Citizen (military style). Now I'm sure my next Japanese watch will be an Orient.