Could be a loose flywheel, a loose connecting rod, something disconnected under the flywheel housing, or a loose baffle under the mowing deck.
Get down, Brother, and peek under it. Might see something under there. Could also be a loose mower blade.

Thanks, Rusty. I should have added when I first wrote that the blade has just been replaced (previous one was bent beyond straightening, which may be an important clue as to what's going on). The new blade comes closer than it looks like it should to one rear corner of the deck, but it happens when both blade tips approach that area, and I don't think that would occur if the shaft was bent.

I've done a bit of checking into other possibilties... Nothing loose under the deck. Nothing loose under the flywheel housing. Flywheel shear pins are intact. The magneto is providing spark on my spark checker, but I don't know what the advance is supposed to be -- what I can say is that when the piston reaches at TDC, the magnets are almost exactly 90 degrees past the magneto. The piston is reaching the top of the block (I've taken the head off to look). The valves are operating at the appropriate times and nothing is broken around the valve springs.

Any further insights?

Check the engine mounts bolts they may be loose.

Engine mounting bolts were solid. HOWEVER, in the process of checking them and rotating the flywheel, I realized that there's a small metal piece attached to the engine block that has several tabs on it that interlock with the side of the flywheel housing when it's attached. I saw this piece moving ever so slightly as I turned the flywheel, but I couldn't figure out why based on the fact that the piece isn't attached to anything that moves.

Then it hit me: as the magnets were going by on the flywheel, they pulled this piece closer. And a close inspection of the flywheel magnets showed tiny scrape marks where this piece has been pulled in close enough to make contact with the spinning flywheel. Smacking the piece with a screwdriver did reproduce a sound similar to what I'd been hearing.

So when I get a chance, the next step will be to put everthing back together and see if bending that metal piece out a bit stops the noise. If not, I think the only thing left to do will be to drain the oil, pull the bottom off the motor, and examine the piston rod to see if maybe it's loose where it attaches to the crankshaft, although at this point I'm not sure how it could be since the piston seems to be operating as it should and there are no odd sounds when I spin the engine by hand.

Thanks!

Ches: You may have bent the crankshaft , when you Hit the Object with the mower blade .

@ewalk: agreed on the bent crankshaft possibility, but wouldn't that mean the blade tips would pass a different distance from a given point on the deck? On this beast, the blade tips both seem to be traveling the same orbit.

FWIW, it's somebody else's mower, so I can't take credit for hitting whatever was hit. :-)

Ches: The clearance should be diminished at some point within the rotation path . Obviously also there should be a scrap mark indicating this also , since you are hearing a metallic sound . The odd think (pun intended) is that it appears intermittent ? I think your 1st inspection of the flywheel area may have been the origination of the noise . Have you had a chance to further check this out. Otherwise the Connecting Rod bearing either Small or Big end may be suspect as Rusty has advised . Good luck in the trouble-shooting , let us know what your discover Bro :)

Ches; you're right, if the crank is bent, the blade tips will not both be the same distance from any one point.
Loose bearings in a mower do not make a clanging, bell-like noise - they make a rattling sound.
The noise must be coming from the flywheel as you speculated.

OK, it's all back together, but something is still clanking once the engine starts to catch. Also, the mower will not fully start, which is odd considering that I've inspected and cleaned all the fuel system parts, replaced the spark plug, checked the spark, replaced the gasket behind the air filter housing, etc.

Although the flywheel shear pins are intact, I can't help but wonder if the timing isn't getting screwed up somewhere. One other bit of info that makes me suspect timing: every 10 pulls or so, the starter rope kicks back wickedly and goes flying out of my hand, and I keep thinking maybe there is someplace that there's an ignition spark when it shouldn't be.

I'm seeing references on the exploded engine diagram to a "timing key" and "timing gear" on the crankshaft. Could those be the culprit? Or do those have to do with valve timing rather than ignition timing?

Certainly sounds like your timing is off. Although you previously siad the Flywheel Key was intact it may be twisted enough to cause the kickback situation. As for the internal noise sounds like the loose piece of metal is still the cause .

And the answer is...

The flywheel key was sheared, so the flywheel was about 60 degrees rotated from where it was supposed to be. I think the clanking noise was a combination of the aforementioned metal piece slapping against the flywheel magnets, plus the sound of pinging/knocking resulting from the fuel igniting when the cylinder wasn't ready for it. Once I managed to get the flywheel off and installed a new key, the engine ran fine.

THANK YOU to everyone who chimed in.

Well there you go Dude ! Never too Old to Learn a New Trick Huh lol ? Been there done that Bro Roflmbo , Thanks for the follow up and Happy Ending :)

'Flywheel shear pins are intact'

'The flywheel key was sheared'

I don't know how it wasn't and then it was, but that's exactly what the key is supposed to do - shear, so the flywheel can shed it's energy without bending the crank. That's also why blades are so ductile, in that they'll bend double without cracking. You're very fortunate that the crankshaft wasn't damaged.