1999 Honda Odyssey will not engage Reverse. Engages all forward gears
Park you car on a level surface. Set the parking brake on you car; start the engine and let it get to full operating temperature; pop the hood and while the engine is running, check the transmission fluid by pulling out the transmission dipstick. Wipe off the dipstick, reinsert it into the dipstick tube all the way down, then pull it out again and check the fluid level.
As you check the fluid level, observe the fluids' color; any discoloration, other than a bright red color, means the transmission is going bad, especially if the fluid smells burnt.
If the fluid passes the visual and smell test, make sure the transmission fluid level is within or between the fill and full marks. If you need to add transmission fluid, make sure you use the right type based on the car manufacturer's specs which are located in your car's owner's manual. In absence of one, call your local dealership or parts store, they can easily look it up free of charge. Of course if the level is low, add a little fluid at a time through the dipstick tube while the car is running- checking the fluid level frequently.
Then check the PCM for any DTCs; are you pulling any fault codes?
If there is damage to the bands, clutches, or solenoids, a tear down and rebuild of the transmission will have to be done; and it is costly