Do the ratchets click when pushing the mower forward with the engine off?

I don't hear anything I would describe as ratcheting. I can push it forward and pull it in reverse, with power off, but I cannot make a turn.

Were the keys inserted rounded side down and lubed well/

Frankly I don't remember, but I didn't do anything to grease them. I'll check tomorrow evening and make sure they are inserted round side down.

Any special grease to use?

Basically any good grease, but I like to stay on the lighter side. Those axle-end gears/ratchets have to be cleaned and lubed yearly. You can use the spray can of white lithium grease sold most places, even Wal Mart. Use the straw that comes with it and really dope it up.
The HRR216 is a nice mower. I just finished re-habbing my stepson's. I put new wheels on, including the new-style front wheels with ball bearings, a new blade set, spark plug, changed oil and air filter, cleaned/lubed the rear end and got him a mulch plug and side discharge chute. They got it from his wife's mom who gave up housekeeping and all it had was a bag, no plug or chute. Plus, a general cleaning/fixing-up. It has some hours on it but runs fine. I still don't think it's going to be around but a few more years. Grass had been allowed to build up and the discharge port housing is very close to being rusted out. The job I did on it would have cost way more than its worth at a shop, but when I take it over there I'll just say 'Happy Father's Day'!
BTW, I found a really good mower cleaner - car wheel cleaner! Safe for aluminum, painted surfaces and polymers. Wet down mower, spray the cleaner on, use a soft-bristle brush all over, hose off - you're done.

What is the mower cleaner you are using?

Auto wheel cleaner - just one I had hanging around, but it's Armor-All 'Extreme' for coated, painted and uncoated wheels. I didn't mention that it's also an aluminum brightener. Don't use it on a hot mower - cold only, and spray it down first. Don't let it dry before rinsing off. It's best to brush before rinsing, but if you don't have a long-handled soft-bristle wheel brush, you can just rinse it off after a few minutes.

You definetly need to lube the ratcheting gears, I had the same problem and fixed it on my Harmony.

Also see this thread "Honda HRB216TDA Transmission Post Mortem" for good pictures of how to reassemble the gears and keys with the correct orientation. I used some disc brake grease that I had on hand and used liberally. Plan to clean/regrease at least every couple of years, its amazing how much easier the mower moves around now.

Thanks for suggesting the "Tranny Post Mortem" thread. However, in trying to understand the difference between the two photos of the ratcheting drive gears I struggle to understand why one is marked wrong vs the other marked right. Can you tell me what is being pointed out that is different between the two?

Life is good, it now turns. It was the keys, I had them in bass ackwards. Thanks to everyone for the suggestions and help.

Another winner for the team!

My HRR216 quadracut mower , the transmission will not pull the unit forward at all , by lifting the drive wheels off the ground the wheels now are turning in all 3 gears any suggestion thanks mike

I had the same problem with rear wheels not making the turns, sure am glad I found this thread, put the keyways in rounded down and no more problems!