Your reverse hub has split open. You'll probably find that it reverses fine on level ground correct? This is because it requires less hydraulic pressure for the gearbox to move the car on level ground, however once trying to reverse up an incline, the hydraulic pressure required is too much for the hub to deal with so oil leaks past the crack in your hub, bleeding off pressure which then releases pressure from the piston in turn releasing pressure on the clutches inside the hub, giving you the loss of drive.
The split hub is caused from people not stopping the car fully when engaging reverse, meaning the clutches in the reverse hub baulk up together and clash. Too much of this will eventually snap the alloy hub casing open which is what has happened here, which is why you are always meant to stop the car fully before going into reverse. Unfortunately, no amount of oil changes, snake oil remedies or magic potions will fix your gearbox. This is a MECHANICAL fault which requires a stripdown to fix.
As for more bad news...If you ever decided to go down the route of getting a second hand gearbox, there is a couple of issues. Firstly, each and every automatic gearbox for these accords are coded to the car. In other words, you could put in a perfectly working auto from another accord and it would not work, as it will realise it's in a different car and force itself into emergency mode the whole time. Also, you can buy a second hand gearbox but find its faulty anyway, as there's no way of testing an automatic outside of the car like you can a manual, unless you have access to 100k worth of hydraulic transmission testing machines!
So all of that means you are only left with one choice, repair the gearbox that is currently I'm the car...and this is where news will only get worse. Naturally, the gearbox will require a full strip down and even though it's just the hub and clutches at fault, there's a few other parts that you MUST change when the gearbox is stripped, so lets talk you through that.
Reverse Hub: £428
Reverse Clutch Pack: £21
Overhaul Kit (needed to reseal the gearbox once opened): £176
Filter: £30
Labour (3.5 Hours at £110 per hour): £385
Total: £1040, and that's before you even take into account the fitting of the gearbox, or the oil etc.
So to summarise, it's probably best you think about changing the car as any sort of lasting repair to your gearbox is going to cost as much as it's worth.
Best of luck with whatever route you choose to take.
www.wexfordtransmission.ie