I did some research in the FSM, which is 337 pages just for the transmission.
I strongly suspect either a bad solenoid and/or an obstruction in one of the valve body passages. There's also the possibility that the countershaft is damaged or that the 5th clutch pack is damaged.
Given that it sometimes goes into reverse and sometimes doesn't, it leads me to think that it's an intermittent solenoid or plugged up passage.
One of the diagnostic steps listed is to drain and strain the fluid. If you see lots of debris, the remedy is to replace the transmission. There are a whole host of diagnostics that can be done with the Honda diagnostic system, but your dealer will never spend the time to confirm the problem. if you take it to a dealer, they will almost certainly tell you that a new transmission is needed.
If the van has a good many miles on it and you've never changed the fluid, you can bet it's gummed up and you'll be hard pressed to get someone to clean all of the components good enough to get a long term fix out of it. You could try several drain and fills and also make sure to replace the filter (yes, there is a filter).
A small transmission specialist may be your best bet. You want to find someone who is willing to diagnose the problem and not just read OBD codes.
