The computer can prevent you from shifting into reverse if it's not getting correct results from the pulse generators and inhibitor switch.
Just for kicks, make sure that everybody is waiting for a number of seconds with the shifter in R - it is possible that it will shift, just not immediately. That doesn't make things better, it just lets you move your car.
You should theoretically check your shift linkage, but since reverse is between park and neutral, I'm betting that if P and N do what they're supposed to do, then R is too.
If you can't shift into reverse immediately with the solenoids unplugged, you're in pretty bad shape. While it COULD be something as "simple" as replacing the filter or the valve body, it also could be the front clutch or the low-reverse brake, neither of which you can service on the vehicle like the end clutch.
So, you have to make a decision about whether you want to risk any sunk costs on troubleshooting the problem, or if you want to go straight to the transmission shop.
The correct way to begin things is with a pressure gauge. There are marginally cheaper ones available, but I was really pleased with the readability of the high pressure dial on this one:
This is a specialized tool you will very rarely use. I can't say whether or not it will pay for itself, but I'm happy I bought it. After replacing my end clutch bearing, I found that the line pressure was 5PSI too high - hopefully lowering it will extend the life of the new one.
If the front clutch and low/reverse brake pass the pressure test, then the transmission needs to be rebuilt / replaced. If they don't, or if you don't have a pressure gauge, then you can try replacing the filter, and then doing the air checks.
This is going to be time consuming and messy. I'm just going to quickly go over my notes on the procedure, without any diagrams.
CAREFULLY remove the valve body. I found that mine didn't just drop out when unbolted, but that there was some sort of glue the factory had used to hold it in place. I had to gently use a small pry bar on carefully chosen spots, working my way around. (This is after you've removed all nine of the bolts with larger heads, keeping track of where they came from.)
The manual valve slides in and out of the valve body on the driver's side as you move the shift lever. Mine stayed in place when I removed it, I believe I had the shifter in P. If not, it'll just fall on your head and roll around on the ground. Try to keep that from happening. Oh, the valve body is heavy.
Use an air gun with a rubber tip to blow LOW pressure compressed air into the oil holes for the front clutch and low-reverse brake. You should hear them clunk and they should hold pressure.
This tells you if they're leaking, not if they're slipping. If they fail the check, the transmission needs to be rebuilt / replaced. If they pass, the valve body should be disassembled and cleaned / rebuilt / replaced. If you still don't have reverse, the transmission needs to be rebuilt / replaced.
I'd hate for anybody to have to go through $20 of fluid (Valvoline Maxlife), $15 gasket / filter (Duralast), $20 more fluid, a valve body (????) and $20 more fluid, and still end up replacing the transmission, but at each step you're trying to avoid replacing the transmission at all cost. So yeah, it's just not a good situation.