First, I too, have a 2000 Odyssey.
For hitch mounted carriers, there is Yakima, Thule, and Bauer. I was going to get the Yakima when I went to a store that had Bauer (which I'd never heard of before). The shop is Sports Rack Outfitters and they carry hitches, carriers, etc. Their web site:
The carrier I got:
It is actually a 2 bike carrier on a swing arm. Very convenient since you can still open the back with the bikes on the carrier. I got the 2 bike add on. Difference from the Yakima, EACH bike is locked to the carrier - when I got mine, the Yakima model that had the swing arm ... only the OUTSIDE bike locked to the carrier.
Also, the BVG also locks to the hitch so somebody can't walk off with your bikes by taking the carrier off from the hitch.
Note: the thing is HEAVY. I would bottom out the Odyssey with 4 bikes on driveways, so I had to go slow. The solution ... I got a Air Lift 1000 for the Odyssey. Essentially, it is two air bladders that sit in the rear suspension. You can adjust the pressure via two extra valves on each side of the car. I normally run 25 psi. Fully loaded, 4 bikes, Yakima Rocket Box on top, loaded to the hilt for a trip to Mammoth Mountain, the Odyssey looks LEVEL when you look at it sideways. Other vehicles I see (including SUVs) are tilted upwards (back is riding LOW).
If you carry mountain bikes with full suspensions, you may need to get dummy top tubes to get them on this carrier. For that, I went with the Yakima ones; they look, well, better built than the others.
Lastly, the reason I got the BVG was it was cheaper than the Yakima and I liked that the carrier and all bikes were locked.