OK - the Beijing Traffic Management Branch (TMB) police have a nifty new email system. You can email them any question, in English, and they will provide a quick response. What a great system! Thanks guys.

Anyways, I emailed it and asked the question about where an "A" plate can go, and whether displacement matters, or whether sidecars (3-wheeled bikes) are exempt. Here is the response:

Dear Richard,

The Jing A plate motorcycle(including a large motorcycle such as a BMW and sidecar motorcycles (3 wheeled)which you mentioned in your email) is not permitted to ride on the 2nd, 3rd and 4th Ring Roads. And it is also prohibited to ride on the Changan street between Xinxing bridge and Guomao bridge from 7:00am to 20:00pm. There are some other places where motorcycle maybe also not allowed to ride, if that happens please check the traffic sign at spot.
Motorcycle does restricted by the plate number in rush hours(7:00am to 20:00pm) on working days(from Monday to friday).
Best wishes
Beijing Traffic Management Bureau


So, there you have it:

1) Can't ride on ChangAn between 0700 and 2000 between Guomao & Xinxing (hmmm, further than I thought)
2) Can't ride on 2nd, 3rd or 4th ring road
3) Sidecars are just like motorcycles (no exemption)
4) Also have to watch the plate number restrictions (e.g., plates ending in 3 or 8 can't ride on Tuesdays)
5) May also be other signed restrictions (e.g., the shiny new 5th Ring Road signs that are popping up in some locations)

That email system is cool.

http://www.bjjtgl.gov.cn/publish/portal1/tab765/

Cheers
Richard